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In algebra, equations of lines can take many forms, but one of the most useful is called the slope-intercept form. When you look at this form on a graph, you slope the line and its y-intercept.
The slope-intercept form is y = mx + b. The m represents the slope of the line. The b is the y-coordinate of the intercept where the line crosses the y-axis. A line with the equation y = –3x + 2 has a slope of –3 and a y-intercept of — that is, the line crosses the y-axis at — (0, 2).
Having the equation of a line in this form makes graphing the line a snap. Follow these steps:
1. Plot the y-intercept on the y-axis.
In this example, y = –3x + 2, mark the y-intercept at (0, 2).
2. Write the slope as a fraction. If the slope is negative, put the negative part in the numerator.
Using the equation y = –3x + 2, the fraction would be –3/1.
The slope has the change in y in the numerator and the change in x in the denominator.
3. Starting at the y-intercept, count the amount of the change in x to the right of the intercept, and then count up or down (depending on whether it's positive or negative) from that point.
Wherever you end up is another point on the line. In this example, you go to the right 1 unit and down (because it's negative) 3 units to land at (1, –1). This is another point in the line defined by y = –3x + 2.
4. Mark that point and draw a line through the two points.
Graphing lines using the slope-intercept form is a piece of cake. But what if the equation isn't in that form? Are you stuck with substituting in values and finding coordinates of points that work? Not necessarily. Changing the form of the equation using algebraic manipulations — and then graphing using the new form — is often easier.
To change the equation of a line to the slope-intercept form, y = mx + b, first isolate the term with y in it on one side of the equation, and then divide each side by any coefficient of y. You can rearrange the terms so the x term, with the slope multiplier, comes first.
For example, to change the equation
4x – 2y = 8
to slope-intercept form, follow these steps:
1. Begin isolating the y by subtracting 4x from each side of the equation. You end up with
–2y = 8 – 4x
2. Next, divide each side by –2. You get this:
y = –4 + 2x
3. Finally, rearrange the equation into the familiar slope-intercept form:
y = 2x – 4
Now graphing this line is a piece of cake.
Now try it out on your own. Put the following equations in slope-intercept form and graph the lines they define:
y = 5x – 2
4x + 2y = 12
4x – y – 3 = 0
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