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Concept Version 14
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Slope-Intercept Equations

The slope-intercept form of a line summarizes the information necessary to quickly construct its graph.

Learning Objective

  • Convert linear equations to slope-intercept form and explain why it is useful


Key Points

    • The slope-intercept form of a line is given by y=mx+by = mx + by=mx+b where mmm is the slope of the line and bbb is the yyy-intercept. 
    • The constant bbb is known as the yyy-intercept.  From slope-intercept form, when x=0x=0x=0, y=by=by=b, and the point (0,b)(0,b)(0,b) is the unique point on the line also on the yyy-axis.
    • To graph a line in slope-intercept form, first plot the yyy-intercept, then use the value of the slope to locate a second point on the line.  If the value of the slope is an integer, use a 111 for the denominator.
    • Use algebra to solve for yyy if the equation is not written in slope-intercept form. Only then can the value of the slope and yyy-intercept be located from the equation accurately.  

Terms

  • slope

    The ratio of the vertical and horizontal distances between two points on a line; zero if the line is horizontal, undefined if it is vertical.

  • y-intercept

    A point at which a line crosses the yyy-axis of a Cartesian grid.


Full Text

Slope-Intercept Form

One of the most common representations for a line is with the slope-intercept form. Such an equation is given by y=mx+by=mx+by=mx+b, where xxx and yyy are variables and mmm and bbb are constants.  When written in this form, the constant mmm is the value of the slope and bbb is the yyy-intercept.  Note that if mmm is 000, then y=by=by=b represents a horizontal line. Note that this equation does not allow for vertical lines, since that would require that mmm be infinite (undefined).  However, a vertical line is defined by the equation x=cx=cx=c for some constant ccc.

Converting an Equation to Slope-Intercept Form

Writing an equation in slope-intercept form is valuable since from the form it is easy to identify the slope and yyy-intercept.  This assists in finding solutions to various problems, such as graphing, comparing two lines to determine if they are parallel or perpendicular and solving a system of equations.

Example

Let's write an equation in slope-intercept form with m=−23m=-\frac{2}{3}m=−​3​​2​​, and b=3b=3b=3.  Simply substitute the values into the slope-intercept form to obtain:

y=−23x+3\displaystyle y=-\frac{2}{3}x+3y=−​3​​2​​x+3

If an equation is not in slope-intercept form, solve for yyy and rewrite the equation.

Example

Let's write the equation 3x+2y=−43x+2y=-43x+2y=−4 in slope-intercept form and identify the slope and yyy-intercept. To solve the equation for yyy, first subtract 3x3x3x from both sides of the equation to get:

2y=−3x−4\displaystyle 2y=-3x-42y=−3x−4

Then divide both sides of the equation by 222 to obtain: 

y=12(−3x−4)\displaystyle y=\frac{1}{2}(-3x-4)y=​2​​1​​(−3x−4)

Which simplifies to y=−32x−2y=-\frac{3}{2}x-2y=−​2​​3​​x−2.  Now that the equation is in slope-intercept form, we see that the slope m=−32m=-\frac{3}{2}m=−​2​​3​​, and the yyy-intercept b=−2b=-2b=−2.

Graphing an Equation in Slope-Intercept Form

We begin by constructing the graph of the equation in the previous example.

Example

We construct the graph the line y=−32x−2y=-\frac{3}{2}x-2y=−​2​​3​​x−2 using the slope-intercept method. We begin by plotting the yyy-intercept b=−2b=-2b=−2, whose coordinates are (0,−2)(0,-2)(0,−2).  The value of the slope dictates where to place the next point.  

Since the value of the slope is −32\frac{-3}{2}​2​​−3​​, the rise is −3-3−3 and the run is 222.  This means that from the yyy-intercept, (0,−2)(0,-2)(0,−2), move 333 units down, and move 222 units right.  Thus we arrive at the point (2,−5)(2,-5)(2,−5) on the line. If the negative sign is placed with the denominator instead the slope would be written as 3−2\frac{3}{-2}​−2​​3​​, we can instead move up 333 units and left 222 units from the yyy-intercept to arrive at the point (−2,1)(-2,1)(−2,1), also on the line.

Slope-intercept graph

Graph of the line y=−32x−2y=-\frac{3}{2}x-2y=−​2​​3​​x−2.

Example

Let's graph the equation 12x−6y−6=012x-6y-6=012x−6y−6=0. First we solve the equation for yyy by subtracting 12x12x12x to obtain: 

−6y−6=−12x\displaystyle -6y-6=-12x−6y−6=−12x   

Next, add 666 to get:

 −6y=−12x+6\displaystyle -6y=-12x+6−6y=−12x+6

Finally, divide all terms by −6-6−6 to get the slope-intercept form:

y=2x−1\displaystyle y=2x-1y=2x−1  

The slope is 222, and the yyy-intercept is −1-1−1.  Using this information, graphing is easy.  Start by plotting the yyy-intercept (0,−1)(0,-1)(0,−1), then use the value of the slope, 21\frac{2}{1}​1​​2​​, to move up 222 units and right 111 unit.

Slope-intercept graph

Graph of the line y=2x−1y=2x-1y=2x−1. 

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