How to Calculate Slope and Intercepts of a Line

The slope of a line measures how steep the line is. http://www.mathopenref.com/coordslope.html You could also say it is the rise over the run; that is, how much the line rises vertically compared with how much it runs horizontally. Being...

Method 1 of 4:

Using a Graph to Find the Slope

Pick two points on the line. Draw dots on the graph to represent these points, and note their coordinates.
1. Remember when graphing points to list the x-coordinate first, then the y-coordinate.
2. For example, you might choose the points (-3, -2) and (5, 4).
Determine the rise between the two points. To do this, you must compare the difference in y of the two points. Begin with the first point, the point that is the farthest left on the graph, and count up until you reach the y-coordinate of the second point.
1. The rise can be positive or negative; that is, you can count up or down to find it.^[4] If the line is moving up and to the right, the rise is positive. If the line is moving down and to the right, the rise is negative. ^[5]
2. For example, if the y-coordinate of the first point is (-2), and the y-coordinate of the second point is (4), you will count up 6 points, so your rise is 6.
Determine the run between the two points. To do this, you must compare the difference in x of the two points. Begin with the first point, the point that is farthest left on the graph, and count over until you reach the x-coordinate of the second point.
1. To run is always positive; that is, you can only count from left to right, never right to left. ^[6]
2. For example, if the x-coordinate of the first point is (-3), and the x-coordinate of the second point is (5), you will count over 8, so your run is 8.
Make a ratio using the rise over the run to determine the slope. The slope is usually in fraction form, but it can also be a whole number.
1. For example, if the rise is 6 and the run is 8, then your slope is $68{displaystyle {frac {6}{8}}} How to Calculate Slope and Intercepts of a Line Picture 1$ , which can be simplified to $34{displaystyle {frac {3}{4}}} How to Calculate Slope and Intercepts of a Line Picture 2$ .

Method 2 of 4:

Using Two Given Points to Find the Slope

Set up the formula $m=y2−y1x2−x1{displaystyle m={frac {y_{2}-y_{1}}{x_{2}-x_{1}}}} How to Calculate Slope and Intercepts of a Line Picture 3$ . In the formula, m = the slope, (x1,y1){displaystyle (x_{1},y_{1})} How to Calculate Slope and Intercepts of a Line Picture 4 = the coordinates of the first point, (x2,y2){displaystyle (x_{2},y_{2})} How to Calculate Slope and Intercepts of a Line Picture 5 = the coordinates of the second point.
1. Remember that the slope is equal to $riserun{displaystyle {frac {rise}{run}}} How to Calculate Slope and Intercepts of a Line Picture 6$ . You are using this formula to find the change in y (rise) over the change in x (run). ^[7]
Plug the x- and y-coordinates into the formula. Make sure you place the coordinates of the first point ((x1,y1){displaystyle (x_{1},y_{1})} How to Calculate Slope and Intercepts of a Line Picture 7 ) and the second point ((x2,y2){displaystyle (x_{2},y_{2})} How to Calculate Slope and Intercepts of a Line Picture 8 ) in the correct positions in the formula, or else you will not calculate the correct slope.
1. For example, given the points (-3, -2) and (5, 4), your formula will look like this: $m=4−(−2)5−(−3){displaystyle m={frac {4-(-2)}{5-(-3)}}} How to Calculate Slope and Intercepts of a Line Picture 9$ .
Complete the calculation and simplify, if possible. This will give you the slope as a fraction or whole number.
1. For example, if your slope is $m=4−(−2)5−(−3){displaystyle m={frac {4-(-2)}{5-(-3)}}} How to Calculate Slope and Intercepts of a Line Picture 10$ you should calculate $4-(-2)=6{displaystyle 4-(-2)=6} How to Calculate Slope and Intercepts of a Line Picture 11$ in the numerator (Remember when subtracting a negative number, you add.) and $5-(-3)=8{displaystyle 5-(-3)=8} How to Calculate Slope and Intercepts of a Line Picture 12$ in the denominator. You can simplify $68{displaystyle {frac {6}{8}}} How to Calculate Slope and Intercepts of a Line Picture 13$ to $34{displaystyle {frac {3}{4}}} How to Calculate Slope and Intercepts of a Line Picture 14$ , so $m=34{displaystyle m={frac {3}{4}}} How to Calculate Slope and Intercepts of a Line Picture 15$ .

Method 3 of 4:

Finding the y-intercept, Given the Slope and One Point

Set up the formula $y=mx+b {displaystyle y=mx+b} How to Calculate Slope and Intercepts of a Line Picture 16$ . In the formula, y = the y-coordinate of any point on the line, m = slope, x = the x-coordinate of any point on the line, and b = the y-intercept.
1. $y=mx+b{displaystyle y=mx+b} How to Calculate Slope and Intercepts of a Line Picture 17$ is the equation of a line. ^[8]
2. The y-intercept is the point at which the line crosses the y-axis.
EXPERT TIP

How to Calculate Slope and Intercepts of a Line Picture 18

Grace Imson, MA
Math Instructor, City College of San Francisco
Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University.

How to Calculate Slope and Intercepts of a Line Picture 19

Grace Imson, MA
Math Instructor, City College of San Francisco

Our Expert Agrees: If you have the slope and one point, plug them into the equation of the line. In y = mx + b, m is the slope, and the point coordinate will contain both x and y. Then, solve for b to find the y-intercept.
Plug in the slope and the coordinates of one point in the line. Remember, the slope is equal to the rise over the run. If you need help finding the slope, see the instructions above.
1. For example, if the slope is $34{displaystyle {frac {3}{4}}} How to Calculate Slope and Intercepts of a Line Picture 20$ , and on point on the line is (5,4), then the formula will look like this: $4=34(5)+b{displaystyle 4={frac {3}{4}}(5)+b} How to Calculate Slope and Intercepts of a Line Picture 21$ .
Complete the equation, solving for b. First multiply the slope and the x-coordinate. Subtract this number from both sides to solve for b.
1. In the example problem the equation becomes $4=334+b{displaystyle 4=3{frac {3}{4}}+b} How to Calculate Slope and Intercepts of a Line Picture 22$ . Subtracting $334{displaystyle 3{frac {3}{4}}} How to Calculate Slope and Intercepts of a Line Picture 23$ from both sides, you end up with $14=b{displaystyle {frac {1}{4}}=b} How to Calculate Slope and Intercepts of a Line Picture 24$ . So the y-intercept is $14{displaystyle {frac {1}{4}}} How to Calculate Slope and Intercepts of a Line Picture 25$ .
Check your work. On a coordinate graph, plot your known point, then draw a line using the slope. To find the y-intercept, look for the point where the line crosses the y-axis.
1. For example, if the slope is $34{displaystyle {frac {3}{4}}} How to Calculate Slope and Intercepts of a Line Picture 26$ , and one point is (5,4), draw a point at (5,4), then draw other points along the line by counting to the left 3 and down 4. When you draw a line through the points, you should see the line cross the y-axis just above the (0,0) coordinate.

Method 4 of 4:

Finding the x-intercept, Given the Slope and Y-intercept

Set up the formula $y=mx+b {displaystyle y=mx+b} How to Calculate Slope and Intercepts of a Line Picture 27$ . In the formula, y = the y-coordinate of any point on the line, m = slope, x = the x-coordinate of any point on the line, and b = the y-intercept.
1. $y=mx+b{displaystyle y=mx+b} How to Calculate Slope and Intercepts of a Line Picture 28$ is the equation of a line. ^[9]
2. The x-intercept is the point at which the line crosses the x-axis.
Plug the slope and y-intercept into the formula. Remember, the slope is equal to the rise over the run. If you need help finding the slope, see the instructions above.
1. For example, if the slope is $34{displaystyle {frac {3}{4}}} How to Calculate Slope and Intercepts of a Line Picture 29$ , and the y-intercept is $14{displaystyle {frac {1}{4}}} How to Calculate Slope and Intercepts of a Line Picture 30$ , the formula will look like this: $y=34x+14{displaystyle y={frac {3}{4}}x+{frac {1}{4}}} How to Calculate Slope and Intercepts of a Line Picture 31$ .
Set y to 0. ^[10] You are looking for the x-intercept, the point at which the line crosses the x-axis. At this point, the y-coordinate will equal zero. So if we set y to 0, and solve for the corresponding x-coordinate, we will find the point (x, 0), which will be the x-intercept.
1. In the example problem, the equation becomes $0=34x+14{displaystyle 0={frac {3}{4}}x+{frac {1}{4}}} How to Calculate Slope and Intercepts of a Line Picture 32$ .
Complete the equation, solving for x. First subtract the y-intercept from both sides. Then divide both sides by the slope.
1. In the example problem the equation becomes $−14=34x{displaystyle {frac {-1}{4}}={frac {3}{4}}x} How to Calculate Slope and Intercepts of a Line Picture 33$ . Dividing both sides by $34{displaystyle {frac {3}{4}}} How to Calculate Slope and Intercepts of a Line Picture 34$ , you end up with $−412=x{displaystyle {frac {-4}{12}}=x} How to Calculate Slope and Intercepts of a Line Picture 35$ . This simplifies to $−13=x{displaystyle {frac {-1}{3}}=x} How to Calculate Slope and Intercepts of a Line Picture 36$ . So the point at which the line crosses the x-axis is $(−13,0){displaystyle ({frac {-1}{3}},0)} How to Calculate Slope and Intercepts of a Line Picture 37$ . So the x-intercept is $−13{displaystyle {frac {-1}{3}}} How to Calculate Slope and Intercepts of a Line Picture 38$ .
Check your work. On a coordinate graph, plot your y-intercept, then draw a line using the slope. To find the x-intercept, look for the point where the line crosses the x-axis.
1. For example, if the slope is $34{displaystyle {frac {3}{4}}} How to Calculate Slope and Intercepts of a Line Picture 39$ , and the y-intercept is $(0,14){displaystyle (0,{frac {1}{4}})} How to Calculate Slope and Intercepts of a Line Picture 40$ , draw a point at $(0,14){displaystyle (0,{frac {1}{4}})} How to Calculate Slope and Intercepts of a Line Picture 41$ , then draw other points along the line by counting to the left 3 and down 4, and to the right 3 and up 4. When you draw a line through the points, you should see the line cross the x-axis just left of the (0,0) coordinate.
How to Calculate Slope and Intercepts of a Line Picture 42

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