What is Slope Intercept Form?

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Linear equations come in three basic forms: point-slope, standard and slope-intercept. The general format of slope-intercept is ​y​ = ​Ax​ + ​B​, where ​A​ and ​B​ are constants. Although the different forms are equivalent, providing the same results, the slope-intercept form quickly gives you valuable information about the line it produces.

TL;DR (Too Long; Didn't Read)

TL;DR (Too Long; Didn't Read)

The slope-intercept form of a line is ​y​ = ​Ax​ + ​B​, where ​A​ and ​B​ are constants and ​x​ and ​y​ are variables.

Slope-Intercept Breakdown

The slope-intercept form, ​y​ = ​Ax​ + ​B​ has two constants, ​A​ and ​B​, and two variables, ​y​ and ​x​. Mathematicians call ​y​ the dependent variable because its value depends on what happens on the other side of the equation. The ​x​ is the independent variable because the rest of the equation depends on it. The constant ​A​ determines the slope of the line and ​B​ is the value of the ​y​-intercept.

Slope and Intercept Defined

The slope of a line reflects the line’s “steepness,” and if it increases or decreases. To give some examples, a horizontal line has a slope of zero, a gently rising line has a slope with a small numeric value, and a steeply rising line has a slope with a large value. The fourth type of slope is undefined; it is vertical. The sign of the slope shows whether the line rises or falls in value going from left to right. A positive slope means the line rises, and a negative slope means it falls.

The intercept is the point at which the line crosses the ​y​-axis. Going back to the form, ​y​ = ​Ax​ + ​B​, you can find the point by taking the value of ​B​ and finding that number on the ​y​ axis, where ​x​ is zero. For example, if your line equation is ​y​ = 2​x​ + 5, the point lies at (0, 5), right on the ​y​ axis.

Two Other Forms

In addition to the slope-intercept form, two other forms are in common use, standard and point-slope. The standard form of a line is ​Ax​ + ​By​ = ​C​, where ​A​, ​B​ and ​C​ are constants. For example, 10​x​ + 2​y​ = 1 describes a line in this form. The point-slope form is ​y​ − ​A​ = ​B​(​x −​ ​C​). This equation provides an example of the point slope form:

y - 2 = 5(x - 7)

Graphing with Slope-Intercept

You need two points to draw a line on a graph. The slope-intercept form gives you one of those points automatically — the intercept. Plot the first point using the value of ​B​ following the directions described above. Finding the second point takes a little algebra work. In your line equation, set the value of ​y​ to zero, then solve for ​x​. For example, using

y = 2x + 5

solve 0 = 2​x​ + 5 for ​x​:

Subtracting 5 from both sides gives you

-5 = 2x

Dividing both sides by 2 gives you

\frac{-5}{2} = x

Mark the point at ( −5/2, 0). You already have a point at (0, 5). Using a ruler, draw a line connecting the two points.

Finding Parallel Lines

Creating a line parallel to one written as slope-intercept is simple. Parallel lines have the same slope but different ​y​-intercepts. So simply keep the slope variable ​A​ from your original line equation and use a different variable for ​B​. For example, to find a line parallel to

y = 3.5x + 20

keep 3.5​x​ and use a different number for ​B​, such as 14, so the equation for the parallel line is

y = 3.5x + 14

You may also need to find a line that passes through a particular point at (​x​, ​y​). For this exercise, plug in the values of ​x​ and ​y​ and solve for the ​y​-intercept, ​B​. For example, you want to find the line that passes through the point (1, 1). Set ​x​ and ​y​ to the values of the point given and solve for ​B​:

Substitute the point values for ​x​ and ​y​:

1 = 3.5 × 1 + B

Multiply the ​x​ value (1) by the slope (3.5):

1 = 3.5 + B

Subtract 3.5 from both sides:

1 - 3.5 = B \\ -2.5 = B

Plug the value of ​B​ into your new equation.

y = 3.5x - 2.5

Finding Perpendicular Lines

Perpendicular lines cross one another at right angles. To do that, the slope of of the perpendicular line is −1 / ​A​ of the original line, or negative one divided by the original slope. To find a line perpendicular to

y = 3.5x + 20

divide −1 by 3.5 and get the result, −2/7. Any line with the slope of −2/7 will be perpendicular to ​y​ = 3.5​x​ + 20. To find a perpendicular line that passes through a given point (​x​, ​y​), plug the values of ​x​ and ​y​ into your equation and solve for the ​y​-intercept, ​B​, as above.

References

About the Author

Chicago native John Papiewski has a physics degree and has been writing since 1991. He has contributed to "Foresight Update," a nanotechnology newsletter from the Foresight Institute. He also contributed to the book, "Nanotechnology: Molecular Speculations on Global Abundance." Please, no workplace calls/emails!