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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; that is, if you 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: Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

3x + 2y = 8
2y = –3x + 8
y = ( ^–3/₂ )x + 4

Then the slope is m =  ^–3/₂ .

There are many contexts in which you will need to solve a linear equation for "y =", so make sure you are comfortable with doing this type of literal equation.

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

Solve for "y =":

2x – y = 5
2x = y + 5
2x – 5 = y

Then y = 2x – 5, so, from the slope-intercept form of y = mx + b, we 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.

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 m =  ¹/₂   and the y-intercept is b =  ^–5/₂ .

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

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 b =  ¹²/₅ .

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

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