GED Mathematical Reasoning: Multi-variable Equations

 

Example 1

Solve for x: ax + by = 3

 

You probably see right off the bat that the equation in example one contains more than one variable. But the variable we are being asked to solve for is “x.” So we will use the concept of opposite operations to get “x” by itself on the left side of the equal sign.

The first thing we will do is isolate the term containing “x” using opposite operations by subtracting the term “by” from both sides. When we do that, the “by’s” cancel from the left side leaving the term “ax.”

On the right side, 3 minus “by” cannot be combined since the terms are not like terms. So we will leave it as just:  3 minus “by”

Now to get “x” by itself, we must “get rid” of the “a” on the left side. Since the “a” and “x” are being multiplied, we will divide by “a” to get “x” by itself.

On the left side, the “a’s” cancel out, leaving just “x.”

On the right side, we have 3 minus “by” all divided by “a.”

Since none of the terms on the right side are like terms, no further simplification is possible. When you’ve gotten the specified variable all by itself on one side of the equal sign, the solving process is complete.

algeb51

 

Example 2

Solve for b: A = \frac{1}{2}bh

 

In this case, we would like to solve for ‘b.’ To do so, we will use the concept of opposite operations to get “b” by itself on one side of the equal sign. Before we do that, though…. Notice that this equation contains the fraction one-half. Let’s multiply both sides of this equation by the Least Common Denominator – or LCD – of two. That will clear the fraction so that we can continue the solving process without it.

Multiplying the left side by 2 leaves two “A.”

On the right side, the 2 and one-half cancel out, leaving just “bh”.

A = \frac{1}{2}bh

(2)A = (2)\frac{1}{2}bh

(2)A = \frac{2}{1}\frac{1}{2}bh

2A = bh

From there, to get “b” by itself on one side of the equal sign, we need to “get rid” of the “h” from the right side. Since the “b” and “h” are multiplied on the right side, we will divide both sides by “h”. That gives us the expression 2 “A” all divided by “h” on the left side. On the right side, the “h’s” cancel, leaving just “b”.

So we have:  2 “A” all divided by “h” equals “b.” Or we can flip things around to read: “b” equals 2”A” all divided by “h”

As we saw in example one, when solving an equation containing more than one variable, it is often the case that expressions cannot be simplified because of the different variables. The expression 2 “A” all divided by “h” cannot be simplified any further.

And when you’ve gotten the specified variable all by itself on one side of the equal sign, the solving process is complete.

algeb52

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