# Algebra: Point-Slope Form

## Point-Slope Form

If you are given the slope of a line and one of the points on the line, then creating the equation of that line is a very simple procedure. All you'll need is the *point-slope formula* for a line.

**Point-slope formula**: If a line has slope *m* and passes through the point (*x*_{1},*y*_{1}), then the equation of the line is

*y*-*y*_{1}=*m*(*x*-*x*_{1})

##### Critical Point

Remember, no matter which of the techniques described in this section you use to find a linear equation, you always need two things: the slope of the line and a point on the line.

##### Critical Point

Variables with subscripts, like *x*_{1} and *y*_{1}, will have completely different values than their non- subscripted look-alikes, *x* and *y*. By the way, that little subscript does not affect the value of the variable at all, like an exponent would. It's just a little garnish that distinguishes between the variables, making them different.

Are you wondering where that *m* came from? For some reason, math people have used the variable *m* to represent the slope of a line for a long time. Believe it or not, no one quite knows why. I could wax historical about this mathematical conundrum, but you'd get bored fast, so let me suffice to say that *m* is the variable used to represent slope in all of the formulas you'll see in this section.

Basically, all you have to do to create a linear equation is to plug in a slope for *m*, an *x*-value from an ordered pair for *x*_{1}, and the matching *y*-value for *y*_{1}, and simplify.

**Example 1**: Write the equation of the line with slope -3 that passes through the point (-1,5) and solve the equation for *y*.

**Solution**: Since the slope equals -3, set *m* = -3 in the point-slope formula. You should also replace *x*_{1} with the known *x*-value (-1) and replace *y*_{1} with the matching *y*-value (5).

*y*-*y*_{1}=*m*(*x*-*x*_{1})*y*- (5) = -3(*x*- (-1))

##### You've Got Problems

Problem 1: Write the equation of the line with slope 4 that passes through the point (2,-7) and solve the equation for *y*.

Simplify the right side of the equation.

*y*- 5 = -3(*x*+ 1)*y*- 5 = -3*x*- 3

Since the problem asks you to solve for *y*, you should isolate it on the left side of the equation by adding 5 to both sides.

*y*= -3*x*+ 2

That's all there is to it! This is the only line in the world that has slope -3 and passes through the point (-1,5).

Excerpted from The Complete Idiot's Guide to Algebra 2004 by W. Michael Kelley. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with **Alpha Books**, a member of Penguin Group (USA) Inc.

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