Solving "Ax + By = C" for "y=" (page 2 of 2)

Sections: Solving for a given variable, Solving for "y="

Probably one of the more important classes of literal equations you will need to solve will be linear equations. For instance, it is common that you are given problems of this type:

• What is the slope of the line with equation 3x + 2y = 8?

In order to find the slope, it is simplest to put this line equation into slope-intercept form. If I rearrange this line to be in the form "y = mx + b", it will be easy to read off the slope m. So I'll solve:

3x + 2y = 8
2y = –3x + 8

y = ( –3/2 ) x + 4

Then the slope is m–3/2.

Warning: There are many contexts, such as graphing and systems of equations, in which you will need to be able to solve a linear equation for "y =", so make sure you are comfortable with these techniques.

• Find the slope and y-intercept of the line with equation 2xy = 5.

2xy = 5
2x = y + 5

2x – 5 = y

Then y = 2x – 5, and, from the slope-intercept form of y = mx + b, I can see that:

the slope is m = 2 and the y-intercept is b = –5.

• Find the slope and y-intercept of the line with equation x – 2y = 5.

I'll solve for "y =":

x – 2y = 5
x = 2y + 5

x – 5 = 2y

( 1/2 ) x – ( 5/2 ) = y

Then y = ( 1/2 ) x – ( 5/2 ), so:

the slope is m1/2  and the y-intercept is b–5/2 .

• Find the slope and y-intercept of the line with equation 4x + 5y = 12.

I'll solve for "y =":

4x + 5y = 12
5y = – 4x + 12

y = ( –4/5 ) x + ( 12/5 )

Then the slope is m–4/5 and the y-intercept is b12/5 .

Don't let literal equations "throw" you. Solving literal equations is just like solving linear (and other sorts of) equations, except that the answers don't simplify as much. The techniques involved are otherwise exactly the same. Just take your time and be sure to write out all your steps clearly.