Slope of a line
In these lessons, we will learn
- how to find the slope of a line from the graph using rise over run.
- how to find the slope of a line using the slope formula.
- how to find the y-intercept from the graph.
Related Pages Calculating Slope More Lessons for Geometry Math Worksheets
The slant of a line is called the slope. Slope describes how steep a line is. The slope of a line can be found using the ratio of rise over run between any two points on the line.
In the following graph, the rise from point P to point Q is 2 and the run from point P to point Q is 4.
Take note that the slope obtained would be the same no matter which two points on the line were selected to determine the rise and the run.
A horizontal line has a slope of zero. A vertical line has an undefined slope.
A line with a positive slope slant upwards, whereas a line with a negative slope slant downwards.
How to find the slope of a line using the ratio of rise over run between any two points on the line?
How to calculate the slope of a line using the rise over run method? It also explains positive slope, negative slope, and the slope of horizontal and vertical lines.
Slope Formula
Slope can also be calculated as the ratio of the change in the y-value over the change in the x-value.
Given any two points on a line, (x 1 , y 1 ) and (x 2 , y 2 ), we can calculate the slope of the line by using this formula:
For example: Given two points, P = (0, –1) and Q = (4,1), on the line we can calculate the slope of the line.
Let’s look at a line that has a negative slope.
For example: Consider the two points, R(–2, 3) and S(0, –1) on the line. What would be the slope of the line?
How to find the slope of the line that passes through two points when given the coordinates of the points? To solve the problem (without graphing), we can use the slope formula, which states that m = (y 2 − y 1 ) / (x 2 − x 1 ). The slope formula can be read as “slope equals the second y-coordinate minus the first y-coordinate over the second x-coordinate minus the first x-coordinate”.
Example: Find the slope of the line passing through the points (5,4) and (8,6)
How to use the slope formula to find the slope of a line given the coordinates of two points on the line? It shows that the slope can be zero or undefined.
- Find the slope of the line containing the points (-10,-4) and (-15,-6)
- Find the slope for (4,8) and (-7,8)
- Find the slope for (6,-9) and (6,-10)
Y-intercept
The y-intercept is where the line intercepts (meets) the y-axis.
In the following diagram, the line intercepts the y-axis at (0,–1). Its y-intercept is equal to –1.
Equation of a straight line can be written in slope-intercept form .
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How to Use the Formula and Calculate Slope
Practice Problems & examples
The slope of a line characterizes the direction of a line . To find the slope, you divide the difference of the y-coordinates of 2 points on a line by the difference of the x-coordinates of those same 2 points .
Different words, same formula
Teachers use different words for the y-coordinates and the the x-coordinates .
- Some call the y-coordinates the rise and the x-coordinates the run .
- Others prefer to use $$ \Delta $$ notation and call the y-coordinates $$ \Delta y$$ , and the x-coordinates the $$ \Delta x$$ .
These words all mean the same thing , which is that the y values are on the top of the formula (numerator) and the x values are on the bottom of the formula (denominator)!
Example One
The slope of a line going through the point (1, 2) and the point (4, 3) is $$ \frac{1}{3}$$.
Remember: difference in the y values goes in the numerator of formula, and the difference in the x values goes in denominator of the formula.
Can either point be $$( x_1 , y_1 ) $$ ?
There is only one way to know!
--> First, we will use point (1, 2) as $$x_1, y_1$$, and as you can see : the slope is $$ \frac{1}{3} $$ . --> Now let's use point (4, 3) as $$x_1, y_1$$, and as you can see , the slope simplifies to the same value: $$ \frac{1}{3} $$ . -->