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9th GRADE MATH PRACTICE PROBLEMS

Problem 1 :

Simplify √20 - √225 + √80

Decompose  20, 225 and 80 into prime factors.

√20  =  √2  ⋅  2  ⋅  5  =  2√5

√225  =  √5  ⋅  5  ⋅  3  ⋅  3  =  5  ⋅  3  =  15

√225  =  √2  ⋅  2  ⋅  2  ⋅  2  ⋅  5  =  (2  ⋅  2)√5  =  4√5

Then, we have

√ 20 -  √ 225 +  √ 80  =   2√5 - 15 + 4√5

√ 20 -  √ 225 +  √ 80  =  6 √5 - 15

√ 20 -  √ 225 +  √ 80   =  6√5 - 15

√ 20 -  √ 225 +  √ 80   =  3(2√5 - 5)

Problem 2 :

The perimeter of the rhombus is 20 cm. One of the diagonals is of length 8 cm. Find the length of the other diagonal.

In a rhombus, the diagonals are intersecting at right angles.

Perimeter of rhombus  =  20 cm

4a  =  20

a  =  5 cm

Length of diagonal  =  8 cm

Half of diagonal  =  4

Let x be the half of the other diagonal.

5 2  =  4 2 + x 2

25  =  16 + x 2

x 2  =  25 - 16

x 2   =  9

x  =  3

Length of other diagonal  =  2(3)

  =  6 cm

Problem 3 :

Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. How much shorter Silvia's trip was, compared to Jerry's trip? 

Let AB  =  1, BC  =  1

Then AC  =   √1 2 + 1 2

AC  =   √2

Distance covered by Jerry  =  1 + 1  ==> 2

Distance covered by Silvia  =   √2

Difference of distance covered  =  2 -  √2

  =  2 - 1.414

  =  0.586

  =  (0.586/ 2)  ⋅ 100%

  =  30%

So Silvia's trip is 30% shorter than the Jerry's trip.

Problem 4 :

Solve log 10 (2x + 50)  =  3

log 10  (2x + 50) = 3

2x + 50  =  10 3

2x  =  1000-50

2x  =  950

x  =  425

So, the value of x is 425.

Problem 5 :

A set has only one element is called ____________ set 

A set has only one element is called as singleton set.

Problem 6 :

Which of the following statement represents this Venn diagram?

In the given venn diagram, the common region for both A and B is not shaded, then the remaining part of B is shaded. So the correct statement is B-A.

Problem 7: 

If (x+p) (x+q) = x 2 -5x-300, find the value of p² + q²

(x+p) (x+q)   =   x 2 -5x-300

x 2 +(p+q)x+pq  =   x 2 -5x-300

p+q  =  -5 and pq  =  -300

p 2 +q 2   =  (p+q) 2 - 2pq

  =  (-5) 2 -2(-300)

  =  25 + 600

p 2 +q 2   =   625

Problem 8 :

The point of concurrence of the angle bisector of a triangle is called the _____________ of the triangle.

The point of concurrence of the angle bisector of a triangle is called the incenter of the triangle.   9

Problem 9 :

The angles are supplementary and larger angle is twice the smaller angle.Find the angles.

Let x be the smaller angle

2x  -  larger angle

2x+x  =  180

3x   180

x  =  60

So, the required angles are 60 and 120.

Problem 10 :

Find the distance between the points

A (-15,-3) and B (7, 1)

Distance between two points  =  √(7+15) 2 + (1+3) 2

  =  √(-22) 2  + 4 2

  =   √(484  + 16)

  =   √500

  =  10 √5

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Free Printable functions Worksheets for 9th Grade

Math functions worksheets for Grade 9 students: Discover a variety of free printable resources to help teachers enhance their students' understanding of mathematical concepts and improve problem-solving skills.

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Explore printable functions worksheets for 9th Grade

Functions worksheets for Grade 9 are essential tools for teachers to help their students master the concepts of Math and algebra. These worksheets provide a variety of exercises and problems that cater to the learning needs of ninth-grade students, focusing on key topics such as linear functions, quadratic functions, and inverse functions. With a range of difficulty levels, these worksheets enable teachers to challenge their students and reinforce their understanding of the subject matter. By incorporating functions worksheets for Grade 9 into their lesson plans, educators can ensure that their students develop a strong foundation in Math and algebra, setting them up for success in future coursework.

Quizizz is a fantastic platform that offers a wide array of resources for teachers, including functions worksheets for Grade 9, Math, and algebra. In addition to worksheets, Quizizz provides interactive quizzes, engaging games, and other educational materials that can be easily integrated into the classroom. Teachers can utilize these resources to create a dynamic and interactive learning environment, keeping their students motivated and interested in the subject matter. Furthermore, Quizizz allows educators to track their students' progress and identify areas where they may need additional support or practice. By incorporating Quizizz into their teaching strategies, teachers can enhance their students' learning experience and ensure that they excel in Grade 9 Math and algebra.

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Ninth Grade (Grade 9) Quadratic Equations and Expressions Questions

You can create printable tests and worksheets from these Grade 9 Quadratic Equations and Expressions questions! Select one or more questions using the checkboxes above each question. Then click the add selected questions to a test button before moving to another page.

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Order of Operations Exercises

Order of operations practice problems with answers.

There are nine (9) problems below that can help you practice your skills in applying the order of operations to simplify numerical expressions. The exercises have varying levels of difficulty which are designed to challenge you to be more extra careful in every step while you apply the rules of the Order of Operations. Good luck!

Part 1: Order of Operations problems involving addition, subtraction, multiplication, and division

Problem 1: Simplify the numerical expression below.

Problem 2: Simplify the numerical expression below.

Problem 3: Simplify the numerical expression below.

Part 2: Order of Operations problems involving the four arithmetic operations and parenthesis (or nested grouping symbols)

Problem 4: Simplify the numerical expression below.

Problem 5: Simplify the numerical expression below.

Problem 6: Simplify the numerical expression below.

Part 3: Order of Operations problems involving the four arithmetic operations, parentheses and exponents

Problem 7: Simplify using the Order of Operations.

Problem 8: Simplify using the Order of Operations.

Problem 9: Simplify using the Order of Operations.

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Using Video Projects to Reinforce Learning in Math

A collaborative project can help students deeply explore math concepts, explain problem-solving strategies, and demonstrate their learning.

Problem-solving, and the creativity that generates and fuels it, lies at the heart of mathematics. Mathematics is essentially about reasoning and much less about memorization or even procedural skills, although both processes are meaningful and useful to simplify and support problem-solving. The National Council of Teachers of Mathematics (NCTM) has consistently advocated to keep problem-solving as the centerpiece of mathematics teaching, and global trends in mathematics education have increasingly emphasized problem-solving and mathematical modeling.

Problem-solving allows students to deepen their conceptual comprehension and appreciate the usefulness and relevance of mathematics. Thus, it generates and fosters interest, engagement, and a deeper understanding of the world around them. Because problem-solving is often used in the mathematics classroom, it’s particularly important to find fresh and interesting ways to attract and maintain students’ engagement.

Video Projects Support Interest in Problem-Solving 

To this end, I assign video projects to my students. In groups of two or three, they solve a set of problems on a topic and then choose one to illustrate, solve, and explain their favorite problem-solving strategy in detail, along with the reasons they chose it. The student-created videos are collected and stored on a Padlet even after I have evaluated them—kept as a reference, keepsake, and support. I have a library of student-created videos that benefit current and future students when they have some difficulties with a topic and associated problems.

Some topics in mathematics are well-suited for applications and problem-solving. These are usually multistep problems that require a combination of strategies and procedural fluency. Typical examples are the motion, work, and mixture problems in algebra, the optimization problems in precalculus or calculus, and related rates problems in calculus.

This collection of student-created videos is about related rates problems (note that some links may not work, as this collection is old). Video activities based on problem-solving can be done at any level of mathematics, as problem-solving is a task in which children are engaged in math class from an early age.

Useful Recording Tools

Some examples of useful recording apps include Screencastify , ScreenPal , iMovie , and QuickTime . Each of them has pros and cons, so I suggest looking at the particular specifications of each tool in terms of the number and length of videos allowed by the free version of those apps. I let my students choose what app they want to use to create their videos—they are generally very familiar with this sort of technology and may be more at ease with one tool over another. All they have to produce is a usable link to their video that will be posted on the common Padlet.

Loom is an intuitive, user-friendly screen recording tool that can record audio, video, browser windows, or entire screens in a Chrome extension, desktop app, or mobile app. You can sign up for a free Loom for Education account; students don’t need an account to watch a teacher’s videos, but they will in order to create their own videos.

Loom’s training module is thorough and includes tutorials, special feature descriptions, and examples. Once you click the Loom icon, there’s a short countdown that precedes the recording. When you stop the recording, a link automatically saves to your clipboard and can be easily shared via email, social media, or an embed code.

The videos will also save to your personal library and can be shared to a team library to make them easily accessible to colleagues. Editing features are quite limited (trimming and changing playback speed), which means you may have to do multiple takes, but teachers can control the settings for comment and download options. 

4 Problem-Solving Strategies

Mathematician George Polya outlined a four-step model in his famous book, How to Solve It . It involves understanding the problem, devising a plan, carrying it out, and finally looking back and reflecting. These are the strategies that my students must demonstrate while creating their videos. 

• Understand the problem: Students reread the problem carefully, summarize and rewrite the information in mathematical notation, use keyword analysis, draw a picture or a diagram, or even act out the scenario.
• Devise a plan: Looking for patterns and solving a simpler problem are my favorite approaches, but other ideas—guess-and-check, working backward, eliminating possibilities, using a formula and solving an equation—can work well too, depending on the circumstances. Most often, for good problems, several of these strategies have to be employed at the same time and help support confidence in the solution.
• Carry out the plan: This is where “show your work” comes in with full force. Communicating their thoughts and ideas is paramount: Students should be systematic, show their thinking in a logical progression, check their work, and be flexible and persistent.
• Look back and reflect: It’s important to consider which part of the problem was the most challenging and why, which process was most effective, and other strategies that could have worked. This makes for more efficient and deeper learning.

Related rates problems can be intimidating at first, and it is useful for students to write out explicitly the steps and strategies they take to solve the first few problems.

My students come up with a model that follows the previously mentioned steps. It includes labeling the rates with their units and sign, an understanding of the rate they must find, finding at least one equation that binds the variables together, differentiating this equation with respect to time, plugging in the given information, and, finally, writing a short sentence that summarizes their conclusion (including sign and units). 

Benefits of the Video Activity

My students and I have experienced several benefits of this task.

Students are encouraged to communicate mathematically. The importance of communication among learners is also heavily emphasized in the NCTM publication Principles and Standards for School Mathematics .

Student collaboration. Viewing learning as a collective endeavor , rather than an individual competition, helps students develop their social and collaborative skills. When students take joint responsibility for their learning—sharing ideas and resources—it fosters a safe environment where they perceive each other as allies rather than competitors, which increases engagement and academic achievement. 

Problem-solving skills are strengthened. As reported in the Executive Summary of the NCTM Principles and Standards for School Mathematics , when solving mathematical problems, students acquire ways of thinking, habits of persistence and curiosity, and confidence in unfamiliar situations that serve them well beyond the classroom. 

Teachers can clearly see students’ understanding. This includes conceptual understanding, procedural precision, logical and analytical thinking, problem-solving strategies, and clarity of communication.

A sense of belonging in math class is cemented. The experience generates positive, affirmative memories—the goal of social and emotional learning—and “ encourages student focus and motivation, improves relationships between students and teachers, and increases student confidence and success .” It should be promoted, particularly in the STEM disciplines.

 In other words, it’s a keeper.

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• What Do 2nd Graders Learn in Math | 9 Key Topics

After building foundational math skills in 1st grade, children move on to 2nd grade, where they learn new math concepts. In 2nd grade, your child will build on what they’ve already learned, delving deeper into numbers, operations, and problem-solving. This year, they’ll strengthen their skills in addition and subtraction and start to explore the basics of multiplication and division. They’ll work with larger numbers, understand place value more thoroughly, and begin to tell time to the nearest five minutes. Your child will also get hands-on with money and basic geometry and start learning about fractions.

Here’s an overview of what 2nd graders typically learn in math and how these concepts contribute to their overall growth.

9 Important Second-Grade Math Topics and Skills 

1. mental addition and subtraction.

In 2nd grade, one of the key math skills your child will develop is mental addition and subtraction. This involves solving basic math problems in their head without needing a pencil and paper. By practicing these skills, your child will become faster and more confident in handling numbers.

At this stage, your child will start with simple addition and subtraction facts, working with numbers up to 20. They’ll gradually move on to larger numbers, learning to add and subtract within 100, and even explore the concept of carrying over (in addition) and borrowing (in subtraction).

To make mental math easier, your child will learn strategies like breaking down numbers into smaller parts and rounding numbers to the nearest ten. For example, instead of adding 47 and 26 all at once, they might break it down to add 40 + 20 and then 7 + 6, combining the sums to get the final answer. And for rounding numbers, your child might round 48 up to 50 or 72 down to 70, making the addition or subtraction simpler.

2. Place Value and Expanded Form

In 2nd grade, your child will continue to build on what they learned about numbers in 1st grade, diving deeper into the concept of place value. This year, they’ll better understand odd and even numbers, working with one-digit, two-digit, and even three-digit numbers. Place value helps them see that each digit in a number has a special role—whether in the ones, tens, or hundreds place.

Your child will practice figuring out which digit stands for ones, tens, or hundreds in various numbers. They’ll also learn how to write numbers in words, making the connection between the digits they see and the words they say.

A big focus this year is on the expanded form . This means breaking down a number to show the value of each digit. For example, 267 would be written as 200+60+7 in expanded form. It’s fun for kids to see how numbers are together and understand what each part represents.

3. Complex Addition and Subtraction

2nd grade math books present various problems that challenge your child to perform more complex addition and subtraction. They’ll also encounter word problems that require them to apply these skills in real-world situations, helping them see the practical side of math.

Your child will be introduced to the number line to make working with larger numbers easier. This tool helps them visualize how numbers move up and down as they add or subtract, making it easier to grasp the concept of working with bigger figures.

4. Coordination, Charts, and Graphs

In 2nd grade, your child will start exploring the world of coordination, charts, and graphs—an important step in developing their data interpretation skills. This year, they’ll be introduced to the basics of graphing on a coordinate grid, learning how to use coordinates to plot images on a graph. It’s a fun and engaging way to help them understand how data can be represented visually.

Through this introduction, your child will get hands-on experience with different types of graphs, including pictographs, bar graphs, and line plots. They’ll learn to read these graphs, interpret the data they show, and even create their own. By practicing with these tools, your child will develop the ability to understand, analyze, and present information clearly and organized.

5. Geometry and Fractions 

In 2nd grade, your child will begin to explore the basics of geometry and fractions, two fundamental concepts that lay the groundwork for more advanced math in later years. They’ll start by learning to identify shapes based on the number of sides they have, which helps them recognize and categorize different types of shapes.

Your child will also be introduced to geometric concepts such as endpoints, line segments, and angles. These ideas help them understand how shapes are formed and how different parts of a shape relate to each other.

As they delve into geometry, your child will also learn about fractions . Using shapes, they’ll discover how to divide a whole into equal parts and how to describe these parts using fractions. They’ll get familiar with terms like numerator and denominator, which are key to understanding how fractions work.

6. Word Problems

In 2nd grade, word problems become a key part of your child’s math learning journey. Nearly every lesson in the 2nd-grade curriculum includes word problems directly related to the topic. These problems are designed to help your child apply the math concepts they’re learning to real-life situations, making the lessons more engaging and practical.

Your child will encounter word problems across various topics, including simple and complex addition and subtraction, the number line, fractions, telling time, and even an introduction to simple multiplication. By working through these problems, your child will develop critical thinking skills and learn how to break down a problem, figure out what’s being asked, and use the appropriate math operations to find the solution.

7. Measurement 

In 2nd grade, your child will start learning about measurement —a key skill that helps them understand the sizes and lengths of objects around them. This year, they’ll be introduced to the ruler, discovering how the lines on one side measure in inches and on the other side in centimeters.

Through hands-on practice, your child will measure images in their books, gaining experience with inches and centimeters. They’ll learn when to use each type of measurement, helping them grasp the concept of different units of measurement and how they apply to various situations.

This measurement lesson teaches your child how to use a ruler and helps them understand the concept of size. It makes it easier for them to compare objects and see the world in a more structured way. By the end of 2nd grade, your child will be more comfortable with measurement, setting the foundation for more advanced math and science concepts.

8. Telling Time

In 2nd grade, your child will begin mastering the important skill of telling time . They’ll start by learning the basics of reading analog and digital clocks, an essential part of their daily routine. Through engaging exercises and clear illustrations, your child will practice identifying the hour hand, minute hand, and second hand on an analog clock.

2nd grade books provide plenty of opportunities for your child to practice telling time, whether reading a clock to the nearest hour, half-hour, or minute. Working with analog and digital clocks, your child will understand how time is measured and represented in different formats.

9. Multiplication

As 2nd grade comes to a close, your child will be introduced to the exciting world of multiplication. This new math operation is a big step forward, and it all starts with learning the times tables. Your child will begin by completing a chart that helps them understand and memorize the times tables for numbers 1 through 12.

The lessons in this section are designed to make multiplication approachable and engaging. Your child will be given plenty of practice to help them master this new skill, building a strong foundation for more advanced math in the future. Through these exercises, they’ll learn how multiplication works, how it relates to addition, and how to use it in real-life situations.

Parental Involvement Tips

Parental support is crucial in helping your 2nd grader succeed in math. The good news is you don’t need to be a math expert to make a big difference! Simple, everyday activities can become fun learning opportunities that show your child how math is used in real life.

• For example, you can start by counting coins together. Whether you’re saving up for something special or just sorting loose change, this activity helps your child practice counting and understanding the value of money.
• Maintaining a family calendar is another great way to involve your child in math. Let them help you mark important dates and count the days of a big event. It’s a fun way to teach them about time and planning.
• When you’re in the kitchen, invite your child to help. Measuring ingredients and following a recipe are perfect ways to practice math skills like fractions and addition. Plus, it’s a great bonding experience!
• Even sorting pantry items can become a math lesson. Ask your child to group items by size, weight, or category. This reinforces their understanding of measurement and classification and helps keep the pantry organized!

By incorporating these simple activities into your daily routine, you’ll make math fun and show your child how math is a part of everyday life. 

End-of-Year Skills Checklist

As your child wraps up 2nd grade, it’s helpful to know what key math skills they should have mastered by the end of the year. Here’s a checklist of the essential abilities your child should be proficient in:

• Adding and Subtracting within 100 : Your child should confidently add and subtract numbers up to 100 mentally and on paper.
• Understanding Place Value : They should be able to break down numbers into hundreds, tens, and ones and understand the value of each digit in a number.
• Mastering Basic Multiplication : By the end of 2nd grade, your child will have a solid introduction to multiplication and should know the times tables for 1 through 12.
• Working with Simple Fractions : Your child should understand basic fractions, like halves, thirds, and quarters, and be able to identify them in shapes and numbers.
• Telling Time : They should be able to read analog and digital clocks, tell time to the nearest five minutes, and understand the concepts of AM and PM.
• Solving Word Problems : Your child should be comfortable solving word problems that involve addition, subtraction, and simple multiplication, applying their math skills to real-life scenarios.
• Understanding and Creating Graphs : Students should know how to read and create simple graphs, such as bar graphs and pictographs, and interpret data using them.
• Measurement Skills : Your child should be able to measure objects using inches and centimeters and understand when to use different units of measurement.
• Identifying Shapes and Geometry Basics : They should be able to recognize and name various shapes, understand basic geometry terms like line segments and angles, and begin to explore 3D shapes.

Supporting Your Child With Second Grade Math At Genie Academy

At Genie Academy, second grade is a crucial time for your child to build a strong foundation in math. That’s why we’re here to help! Our tutoring center supports kids and teens with core subjects like math, abacus , reading , writing , and coding . But let’s be honest—math is one of our favorites, and we’re good at it!

Our expert tutors have a knack for making math fun and understandable. They use creative and engaging methods personalized to your child’s unique learning style so math becomes something they enjoy rather than struggle with. We’ve seen so many second graders flourish in math after working with us, and we’re proud to be a part of their success stories.

Whether you prefer online sessions or in-person tutoring, we’ve got you covered with flexible options that fit your family’s needs. Our curriculum is not just productive—it’s designed to be interactive and enjoyable, making learning a rewarding experience for your child.

We offer both in-person and online classes. Our locations include East Brunswick , Hillsborough , Marlboro , South Brunswick , Plainsboro , and South Plainfield , New Jersey. 

Learn more about us , our core values , our  philosophy , our numerous success stories , and our  history . Also, read our reviews and testimonials and learn about the true benefits of Genie Academy. 

Second grade is a crucial time for your child to build a strong foundation in maths that will prepare them for future success. These skills are essential to boosting their confidence and understanding, from mental arithmetic to mastering basic multiplication.

At Genie Academy, we’re here to make learning maths fun and effective. Our experienced tutors are dedicated to helping your child succeed, and with our flexible online and in-person options, we ensure they get the support they need. We’re excited to be a part of your child’s success in maths and see them grow into a confident prodigy!

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Home » Extras » Class 9 Maths Competency Based Questions 2024-25: Download PDF

Class 9 Maths Competency Based Questions 2024-25: Download PDF

Class 9 Maths Competency Based Questions are available for download here on aglasem.com. These Class 9 Competency Based Questions include MCQs, fill in the blanks, short answer questions, long answer questions, and answer key from the Maths curriculum and need you to apply understanding a little beyond the 9th class Maths textbook. For CBSE students, understanding and mastering these questions is crucial for success in Maths exam. Here you will understand what Competency Based Questions for Maths are, how they differ from traditional questions, and get tips on tackling these questions. You can also download Class 9 Competency Based Questions Maths PDF .

Class 9 Maths Competency Based Questions 2024-25

Here you can access a wide array of CBSE Competency Based Questions for the Maths subject. The PDF includes Multiple Choice Questions, Fill in the Blanks, Short Answer Questions, and Long Answer Questions, along with the answer key for class 9 Maths. The Competency Based Questions for Class 9 Maths can be found here.

Class 9 Maths Competency Focused Practice Questions Download Link – Click Here to Download 9 Maths CBQ PDF

Class 9 Maths Competency Based Questions PDF

The complete pdf for competency focused practice questions for Maths is as follows.

What are Maths Competency Based Questions?

These are designed to evaluate a student’s understanding of concepts of Maths, their ability to apply knowledge in real-life situations, and their critical thinking skills. Unlike traditional questions that may focus on rote learning or memorization, Competency Focused Questions require students to demonstrate a deeper comprehension of the subject matter (Maths). These questions measure various competencies in the Maths subject, such as problem-solving, logical reasoning, creativity, and subject-specific skills.

For Class 9 Competency Based Questions of Maths , students are expected to think beyond the Maths textbook and demonstrate their skills in a way that aligns with real-world applications.

• Class 9 Competency Based Questions

In addition to Maths, the Competency Focused Questions for all subjects for students in 9th standard are as follows.

Other Classes CBQ Download Links

Similarly the class wise CBQ practice question bank for school board are as follows.

• Class 3 Competency Based Questions
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Class 9 Maths Competency Questions  – An Overview

The highlights of this educational resource are as follows.

Competency Based Questions are revolutionizing the way students approach their studies. By focusing on the core competencies required by the CBSE board, students in Class 9 can develop the skills necessary to excel academically and beyond. Whether you are dealing with Class 9 Maths Competency Based Questions or any other subject, remember to focus on understanding, application, and critical thinking.

Start practicing today, and make these questions a core part of your study strategy. The more you engage with these questions, the more confident you’ll become in your ability to tackle any challenge that comes your way!

If you have any queries on 9th Maths Specimen Paper 2025, then please ask in comments below.

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Free Mathematics Tutorials

Solutions to equations - grade 9.

The solutions to the questions in solve equations are presented along with all the steps needed and detailed explanations.

Solutions to Questions in solve equations

• ) Given \( \quad \quad 2x + 2 = 6 \) Subtract \( 2 \) from both sides of the equation \( \quad \quad 2x + 2 - 2 = 6 - 2 \) Group like terms and simplify \( \quad \quad 2x = 4 \) Divide both sides of the equation by \( 2 \) (or multiply both sides by \dfrac{1}{2} \) \( \quad \quad \dfrac{2x}{2} = \dfrac{4}{2} \) Simplify \( \quad \quad x = 2 \) Check the solution obtained Left side of equation for \( x = 2 \) : \( \quad 2x + 2 = 2(2)+2 = 6 \) Left side and right are both equal to \( 6 \) for \( x = 2 \), therefore \( x = 2 \) is a solution to the given equation.
• ) Given the equation \( \quad \quad 5y - 2 = 7y - 8 \) Let us arrange the equation so that the variable is on the right side and the constant is on the left side. Subtract \( 5y \) from both sides. \( \quad \quad 5y - 2 - 5 y = 7y - 8 - 5 y \) Group like terms and simplify \( \quad \quad - 2 = 2y - 8 \) Add \( 8 \) to both sides. \( \quad \quad - 2 + 8 = 2y - 8 + 8 \) Group like terms and simplify \( \quad \quad 6 = 2y \) Divide both sides of the equation by \( 2 \) (or multiply both sides by \( \dfrac{1}{2} \) ). \( \quad \quad \dfrac{6}{2} = \dfrac{2y}{2} \) Simplify \( \quad \quad 3 = y \) Check the solution obtained Left side of equation for \( y = 3 \) : \( \quad 5y - 2 = 5(3) - 2 = 13 \) Right side of equation for \( y = 3 \) : \( \quad 7y - 8 = 7(3) - 8 = 13 \) Hence \( y = 3 \) is a solution to the given equation.
• ) Given \( \quad \quad -2x + 4 + 5x = 7 + 4x - 3 \) Group like terms and simplify \( \quad \quad 3x + 4 = 4 + 4x \) Subtract \( 4 \) and \( 3x \) from both sides. (Two opeartions in one step). \( \quad \quad 3x + 4 - 4 - 3x = 4 + 4x - 4 - 3x \) Group like terms and simplify \( \quad \quad 0 = x \) Check the solution obtained Left side of equation for \( x = 0 \) : \( \quad -2x + 4 + 5x = -2(0) + 4 + 5(0) = 4 \) Right side of equation for \( x = 0 \) : \( \quad 7 + 4x - 3 = 7 + 4(0) - 3 = 4 \) Hence \( x = 0 \) is a solution to the given equation.
• ) Given \( \quad \quad 0.2 d + 4 = - 0.1 d - 2 \) Add \( 0.1d \) to both sides and subtract \( 4 \) from both sides. (Two opeartions in one step). \( \quad \quad 0.2 d + 4 + 0.1 d - 4 = - 0.1 d - 2 + 0.1 d - 4 \) Group like terms and simplify \( \quad \quad 0.3 d = - 6 \) Divide both sides by \( 0.3 \) and simplify \( \quad \quad \dfrac{0.3 d}{0.3} = - \dfrac{6}{0.3} \) \( \quad \quad d = - 20 \) Check the solution obtained Left side of equation for \( d = -20 \) : \( \quad 0.2 d + 4 = 0.2(-20) + 4 = 0 \) Right side of equation for \( d = - 20 \) : \( \quad - 0.1 d - 2 = - 0.1 (-20) - 2 = 0 \) Hence \( d = -20 \) is a solution to the given equation.
• ) Given the equation \( \quad \quad -2(2x- 6) = -(x - 4) \) Use distributive law to expand both sides. \( \quad \quad -4 x + 12 = - x + 4 \) Add \( x \) to both sides, group like terms and simplify \( \quad \quad -4 x + 12 + x = - x + 4 + x \) \( \quad \quad -3 x + 12 = 4 \) Subtract \( 12 \) from both sides, group like terms and simplify \( \quad \quad -3 x + 12 - 12 = 4 - 12 \) \( \quad \quad -3 x = - 8 \) Divide both sides by \( - 3 \) and simplify \( \quad \quad - \dfrac{ 3 x}{-3} = - \dfrac{8}{-3} \) \( \quad \quad x = \dfrac{8}{3} \) Check the solution obtained Left side of equation for \( x = \dfrac{8}{3} \) : \( \quad -2(2x- 6) = -2(2 (\dfrac{8}{3})- 6) = \dfrac{4}{3} \) Right side of equation for \( x = \dfrac{8}{3} \) : \( \quad -(\dfrac{8}{3} - 4) = \dfrac{4}{3} \) Hence \( x = \dfrac{8}{3} \) is a solution to the given equation.
• ) Given \( \quad \quad -(x+2)+4 = 2(x+3) + x \) Use distributive law to expand both sides, group like terms and simplify. \( \quad \quad -x - 2 +4 = 2x + 6 + x \) \( \quad \quad -x + 2 = 3x + 6 \) Add \( x \) to both sides and subtract \( 6 \) from both sides, group like terms and simplify. \( \quad \quad -x + 2 + x - 6 = 3x + 6 + x - 6 \) \(\quad \quad - 4 = 4 x \) Divide both sides by \( 4 \) to obtain the solution \(\quad \quad x = - 1 \) Check the solution obtained Left side of equation for \( x = - 1 \) : \( \quad -(x+2)+4 = -(-1+2)+4 = 3\) Right side of equation for \( x = - 1 \) : \( \quad = 2(x+3) + x = 2(-1+3) - 1 = 3 \) Hence \( x = - 1 \) is a solution to the given equation.
• ) Given \( \quad \quad \dfrac{x}{5} = - 6 \) In order to get rid of the fraction, multiply both sides of the equation by the denominator \( 5 \) \( \quad \quad 5 \left( \dfrac{x}{5} \right) = 5(- 6) \) Rearrange as \( \quad \quad \dfrac{5}{5} x = 5(- 6) \) Simplify to find the solution. \( \quad \quad x = - 30 \) Check the solution obtained Left side of equation for \( x = - 30 \) : \( \quad \dfrac{x}{5} = \dfrac{-30}{5} = - 6 \) Right side of equation is equal to \( - 6 \). Hence \( x = - 30 \) is a solution to the given equation.
• ) Given the equation \( \quad \quad - \dfrac{x}{3} = \dfrac{1}{2} \) Multiply both sides by \( - 3 \) \( \quad \quad - 3 \left(- \dfrac{x}{3} \right) = - 3 \left(\dfrac{1}{2} \right) \) Simplify \( \quad \quad \dfrac{3}{3} x = \dfrac{-3}{2} \) \( \quad \quad x = - \dfrac{3}{2} \) Check the solution obtained Left side of equation for \( x = - \dfrac{3}{2} \) : \( \quad - \dfrac{x}{3} = - \dfrac{1}{3} x = - \dfrac{1}{3} \left( - \dfrac{3}{2} \right) = \dfrac{1}{2} \) Both sides of the equation are equal for \( x = - \dfrac{3}{2} \) Hence \( x = - \dfrac{3}{2} \) is a solution to the given equation.
• ) Given \( \quad \quad - \dfrac{x}{4} = \dfrac{1}{2} - x \) Multiply both sides by the lowest common multiple (LCM) of the denominators \( 4 \) and \( 2 \) which is equal to \( 4 \). \( \quad \quad 4 \left(- \dfrac{x}{4} \right) = 4 \left(\dfrac{1}{2} - x\right) \) Use distributive law to expand the right side of the equation \( \quad \quad 4 \left(- \dfrac{x}{4} \right) = 4 \left(\dfrac{1}{2} \right) - 4 x \) Rearrange the fractions as \( \quad \quad - \dfrac{4}{4} x = \dfrac{4}{2} - 4 x \) Simplify \( \quad \quad - x = 2 - 4 x \) Add \( 4x \) to both sides of the equation and group like terms \( \quad \quad - x + 4 x = 2 - 4 x + 4 x \) \( \quad \quad 3x = 2 \) Divide boh sides by \( 3 \) and simplify to find the solution. \( \quad \quad x = \dfrac{2}{3} \) Check the solution obtained Left side of equation for \( x = \dfrac{2}{3} \) : \( \quad - \dfrac{x}{4} = - \dfrac{1}{4} (x) = - \dfrac{1}{4} \left( \dfrac{2}{3} \right) = - \dfrac{1}{6} \) Right side of equation for \( x = \dfrac{2}{3} \) : \( \quad \dfrac{1}{2} - x = \dfrac{1}{2} - \dfrac{2}{3} = - \dfrac{1}{6} \) Both sides of the equation are equal for \( x = \dfrac{2}{3} \) Hence \( x = \dfrac{2}{3} \) is a solution to the given equation.
• ) Given the equation \( \quad \quad - \dfrac{x-3}{7} = \dfrac{1}{2} (- 2x + 6) \) Find the lowest common multiple (LCM) of the denominators \( 7 \) and \( 2 \) which is equal to \( 14 \) and multiply both sides of the equation by \( 14 \). \( \quad \quad 14 \left(- \dfrac{x-3}{7} \right) = 14 \left(\dfrac{1}{2} (- 2x + 6) \right) \) Rearrange fractions in order to make it easier to simplify \( \quad \quad - \dfrac{14}{7} (x - 3) = \dfrac{14}{2} (- 2x + 6) \) Reduce the fractions \( 14/7 = 2 \) and \( 14/2 = 7\) \( \quad \quad - 2 (x - 3) = 7(- 2x + 6) \) Use distibutive law to expand both sides \( \quad \quad -2x + 6 = - 14 x + 42 \) Subtract \( 6 \) from both sides \( \quad \quad -2x + 6 - 6 = - 14 x + 42 - 6 \) Group like terms and simplify \( \quad \quad - 2 x = - 14 x + 36 \) Add \( 14 x \) to both sides \( \quad \quad - 2 x + 14 x = - 14 x + 36 + 14 x \) Group like terms and simplify \( \quad \quad 12 x = 36 \) Divide both sides by \( 12 \) \( \quad \quad \dfrac{12x}{12} x = \dfrac{36}{12} \) Simplify \( \quad \quad x = 3 \) Check the solution obtained Left side of equation for \( x = 3 \) : \( \quad - \dfrac{x-3}{7} = - \dfrac{3-3}{7} = 0 \) Right side of equation for \( x = 3 \) : \( \quad \dfrac{1}{2} (- 2x + 6) = \dfrac{1}{2} (- 2(3) + 6) = 0 \) Both sides of the equation are equal for \( x = 3 \) Therefore \( x = 3 \) is a solution to the given equation.
• ) Given \( \quad \quad - \dfrac{1}{2} - x + 5 = \dfrac{1}{5} + 2(x-2) \) Find the lowest common multiple (LCM) of the denominators \( 2 \) and \( 5 \) which is equal to \( 10 \) and multiply both sides of the equation by \( 10 \). \( \quad \quad 10 \left(- \dfrac{1}{2} - x + 5 \right) = 10 \left( \dfrac{1}{5} + 2(x-2) \right)\) Use distributive law to exppand both sides of the equation \( \quad \quad - 10 \left( \dfrac{1}{2} \right) -10x + 50 = 10 \left( \dfrac{1}{5} \right) + 20(x - 2) \) Expand \( 20(x - 2) = 2x - 40 \) and simplify the fractions \( 10 \left( \dfrac{1}{2} \right) = 10/2 = 5 \) and \( 10 \left( \dfrac{1}{5} \right) = 10/5 = 2 \) \( \quad \quad -5 - 10x + 50 = 2 + 20 x - 40 \) Group like terms and simplify \( \quad \quad - 10 x + 45 = 20 x - 38 \) Add \( 10 x \) from both sides \( \quad \quad - 10 x + 45 + 10 x = 20 x - 38 + 10 x\) Group like terms and simplify \( \quad \quad 45 = 30x - 38 \) Add \( 38 \) to both sides \( \quad \quad 45 + 38 = 30 x\) Group like terms and simplify \( \quad \quad 83 = 30x \) Divide both sides by \( 30 \) and simplify to solve for \( x \). \( \quad \quad x = \dfrac{83}{30} \) Check the solution obtained Left side of equation for \( x = \dfrac{83}{30} \) : \( \quad - \dfrac{1}{2} - x + 5 = - \dfrac{1}{2} - \dfrac{83}{30} + 5 = 26/15 \) Right side of equation for \( x = \dfrac{83}{30} \) : \( \quad \dfrac{1}{5} + 2(x-2) = \dfrac{1}{5} + 2(\dfrac{83}{30}-2) = 26/15 \) Both sides of the equation are equal for \( x = \dfrac{83}{30} \) Therefore \( x = \dfrac{83}{30} \) is a solution to the given equation.

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