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Writing Linear Equations in Slope-Intercept Form

Objective Learn the slope-intercept form of the equation defining a line.

The slope-intercept form is an extremely useful special case of the point-slope form of the equation defining a straight line. It is written as y = mx + b, where m is the slope and b is the y-intercept, the point where the line intersects the y-axis. The x- and y-intercepts of a line are the points where the line intersects the x- and y-axes, respectively.

First, we will try to answer the question “How can we determine the x- and y-intercepts of a line that is given in the form of an equation?”

Then we will learn how to find the intercepts algebraically by solving linear equations. This can be done whether the equations are in standard form or in point-slope form.

 

Example 1

Find the y-intercept of the graph of 2x + 3y = 12.

Solution

This equation is in standard form. The y-intercept is the intersection of the line and the y-axis, so it is a point that is both on the line and on the y-axis. It satisfies the following equations.

2x + 3y = 12 Equation of the line
x = 0 Equation of the y-axis

To find the y-intercept, let x = 0 in the equation of the line.

2x + 3y = 12  
2(0) + 3y = 12 Let x = 0.
3y = 12  
y = 4 Divide each side by 3.

The y-intercept of this line is 4, so the line crosses the y-axis at (0, 4).

 

Example 2

Find the y-intercept of the graph of 2( y - 1) = 5( x + 2).

Solution

This equation is in point-slope form. As in the case of standard form, let x = 0 and solve for y.

2( y - 1) = 5( x + 2)  
2( y - 1) = 5( 0 + 2) Let x = 0.
2y - 2 = 10 Distributive Property
2y = 12 Add 2 to each side.
y = 6 Divide each side by 2.

So, the y-intercept is 6 and the line crosses the y-axis at (0, 6).

To find the x-intercept, let y = 0 in the equation, since the x-axis is given by the equation y = 0. Practice finding the x and y-intercepts of the graphs of equations.

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