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Slope-Intercept Form

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Updated on Feb 18, 2014

Slope-intercept form is one of the most common ways to express a linear equation. When plotted on the x-y axis, a linear equation creates a straight line.

 

slope-intercept form

 

The best thing about slope-intercept form is that it shows you two vital statistics of the line at a glance: the line's slopem, and its y-intercept at the point (0, b). With this information, you can quickly sketch the line on the x-y axis.

Example 1:      

equation 1
 
slope: m = 5
y-intercept: b = ​-4
 
slope graph 1
 

 

Example 2:

equation 2
 
slope: m = -2
y-intercept: b = 7
 
graph 2

 

Plot Lines Using Slope-Intercept Form

  1. Mark the y-intercept point (0, b).
  2. Using the slope value, find another point on the line. A positive value for slope indicates a line that goes up from left to right. A negative value indicates a line that goes down ​from left to right.
  3. Connect the points and extend the line in both directions.

 

Practice Problems

Problem A:

equation 3
 

This line crosses the y-axis at the point (0, -4) and has a slope of 3.

A slope of 3 means that the line's rise to run ratio is 3:1. In other words, it rises three units for every one unit it runs from left to right.

  1. Mark the point (0, -4).
  2. Extrapolate: from the y-intercept, go up three and over one, and draw another point at (1, -1).
  3. Connect the points, and extend the line.
graph 3
 

 

Problem B:

equation 4
 

The line crosses the y-axis at (0, 25) and has a slope of -5.

In other words, the line "drops" five units for every one unit it goes right.

  1. Mark the y-intercept (0, 25).
  2. Extrapolate: from the y-intercept, go down five and over one, and mark that point (1, 20).
  3. Connect the points and extend the line.
graph 4
 

 

Problem C:

equation 5
 

The coefficient of in this one is -½, which means this line drops down one unit for every two it goes right.

  1. Find the y-intercept at (0, 3).
  2. Extrapolate using the slope of -½. Count one unit down, then two units to the right, and draw a point at (2, 2).
  3. Connect the points and extend.
graph 5

 

How to Put a Linear Equation Into Slope-Intercept Form

Isolating yTo get a linear equation into slope-intercept form, you need to get the variable all alone on one side of the equal sign. Once you isolate y, the coefficient of becomes the constant m, the slope, and whatever's left over is b, your y-intercept (the point where the line crosses the y-axis).

Example 1: 

equation 6
 
  • To isolate y, you have to subtract 3x from both sides.
slope intercept form 1

 

Example 2: 

slope intercept form 2
 
  • To isolate y, first subtract 2from both sides.

 

slope intercept form 2.1
 
  • Divide both sides by -4.
slope intercept form 2.2
 
  • Then, simplify the equation.
slope intercept form 2.3
 

How to Calculate the Slope of a Line When Given Two Points

 

rise over run
 
slope-definition

 

Calculate the slope of a line that connects the points (1, 2) and (5, 4).

 

finding slope graph
 
  • The difference in the values of the y-coordinates (the "rise") of the two points is (4 - 2) = 2.
  • The difference in the values of the x-coordinates (the "run") of the two points is (5 - 1) = 4.

So, slope = rise ÷ run = 2 ÷ 4 = ½.

*When you're calculating slope, it doesn't matter which point you call (x1, y1) and which one you call (x2, y2). You just need to be consistent when plugging in the values into the equation.

Try it!

What are the slopes of the following pairs?

  1. (2, 3) and (7, 4)
  2. (3, 2) and (4, 7)
  3. (-5, 2) and (1, 3)
  4. (3, -5) and (-4, 9)
  5. (-1, -1) and (0, -2)
  6. (4, -9) and (-4, -17)

(Answers: 1/5; 5; 1/6; -2; -1; 1)

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