Slope-Intercept Form (page 2)
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.
The best thing about slope-intercept form is that it shows you two vital statistics of the line at a glance: the line's slope, m, and its y-intercept at the point (0, b). With this information, you can quickly sketch the line on the x-y axis.
Plot Lines Using Slope-Intercept Form
- Mark the y-intercept point (0, b).
- 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.
- Connect the points and extend the line in both directions.
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.
- Mark the point (0, -4).
- Extrapolate: from the y-intercept, go up three and over one, and draw another point at (1, -1).
- Connect the points, and extend the line.
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.
- Mark the y-intercept (0, 25).
- Extrapolate: from the y-intercept, go down five and over one, and mark that point (1, 20).
- Connect the points and extend the line.
The coefficient of x in this one is -½, which means this line drops down one unit for every two it goes right.
- Find the y-intercept at (0, 3).
- Extrapolate using the slope of -½. Count one unit down, then two units to the right, and draw a point at (2, 2).
- Connect the points and extend.
How to Put a Linear Equation Into Slope-Intercept Form
Isolating y: To get a linear equation into slope-intercept form, you need to get the y variable all alone on one side of the equal sign. Once you isolate y, the coefficient of x 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).
- To isolate y, you have to subtract 3x from both sides.
- To isolate y, first subtract 2x from both sides.
- Divide both sides by -4.
- Then, simplify the equation.
How to Calculate the Slope of a Line When Given Two Points
Calculate the slope of a line that connects the points (1, 2) and (5, 4).
- 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.
What are the slopes of the following pairs?
- (2, 3) and (7, 4)
- (3, 2) and (4, 7)
- (-5, 2) and (1, 3)
- (3, -5) and (-4, 9)
- (-1, -1) and (0, -2)
- (4, -9) and (-4, -17)
(Answers: 1/5; 5; 1/6; -2; -1; 1)
How to Find the Equation of a Line When From Two Points
- Calculate the slope of the line.
- Plug the slope into y = mx + b.
- Plug in the x- and y-values from one of the two given points.
- Solve for b.
Find the equation of the line containing points A: (4, 1) and B: (2, 3).
- [3 - (-1)] / (2 - 4) = 4 / -2 = -1.
- y = -x + b
- (3) = -(2) + b
- b = 5
Solution: y = -x + 5
Find the equation of the line containing (⅔, 4) and (-12, -.25).
- [4 - (-.25)] / [⅔ - (-12)] = 4.25 / 12.67 = .335
- y = .335x + b
- (4) = .335 (⅔) + b
- 4 = .224 + b ; b = 3.776
Solution: y = .335x + 3.776
Other Ways to Express a Linear Equation
2x + 5y = 20
a = 2, b = 5, c = 20
-x/2 - 5y/3 = 12
a = -1⁄2, b = -5⁄3, c = 12
An equation of a line with a slope of 2 that intersects the point (4, 5):
y - 5 = 2(x - 4)
An equation of a line with a slope of -2⁄3 that intersects the point (-2, -1):
y - (-1) = (-2⁄3) (x - (-2) )
y + 1 = -2⁄3(x + 2)
David Travis is the founder and CEO of Prospect Prep, a New York City-based tutoring agency dedicated to helping students earn better grades, higher scores, and acceptance letters from the colleges of their dreams.
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