8 Rearranging Equations
You will need to solve mathematical equations where all but one of the variables is given. The most common method for doing this is to rearrange the equation so that the unknown variable is alone as a numerator on one side of the equals sign, with all the other variables and constants on the other side. While solving, remembering the following will be useful.
- What you do to one side of the equation, you must do to the other side.
- Multiplication and division are inverse operations (that is, they undo each other). Therefore, [latex]\frac{ab}{a}[/latex] simplifies to b.
- Addition and subtraction are inverse operations.
- Dividing is the same thing as multiplying by the inverse.
- ab is the same as ba.
- a = b can be written at b = a.
- Anything can be written as a fraction with a denominator of 1. Therefore, [latex]a\frac{b}{c}[/latex] is the same as [latex]\frac{ab}{c}[/latex].
Here are a couple of simple examples, where only multiplication and division is needed to solve for the specified variable.
Example 7.1
Solve [latex]d=\frac{m}{V}[/latex] for m.
Answer: To get rid of the V on the right side, multiply both sides by V
[latex]V*d=\frac{m}{V}*V[/latex]
Canceling the V’s on the right side gives
m = Vd
Example 7.2
Solve PV = nRT for T.
Answer: To get rid of the nR on the right side, divide both sides by nR
[latex]\frac{PV}{nR}=\frac{nRT}{nR}[/latex]
Canceling the nR on the right side gives
[latex]T=\frac{PV}{nR}[/latex]
When the variable you are solving for is in the denominator, be sure to first get it in the numerator, then get it by itself. See the following examples.
Example 7.3
Solve [latex]M=\frac{mol}{L}[/latex] for L.
Answer: First, get L in the numerator by multiplying both sides by L
[latex]L*M=\frac{mol}{L}*L[/latex]
Canceling L on the right side gives
[latex]LM=mol[/latex]
Now, isolate L on the left side by dividing both sides by M
[latex]\frac{LM}{M}=\frac{mol}{M}[/latex]
Canceling M on the left side gives
[latex]L=\frac{mol}{M}[/latex]
Example 7.4
Solve [latex]\frac{V_1}{T_1}=\frac{V_2}{T_2}[/latex] for T2.
Answer: First, get T2 in the numerator by multiplying both sides by T2
[latex]T_2*\frac{V_1}{T_1}=\frac{V_2}{T_2}*T_2[/latex]
Canceling T2 on the right side gives
[latex]\frac{T_2V_1}{T_1}=V_2[/latex]
Now, isolate T2 on the left side by multiplying both sides by T1 and dividing both sides by V1
[latex]\frac{T_1}{V_1}*\frac{T_2V_1}{T_1}=V_2*\frac{T_1}{V_1}[/latex]
Canceling T1 and V1 on the left gives
[latex]T_2=\frac{V_2T_1}{V_1}[/latex]
If the equation has addition or subtraction as well as multiplication and division, be sure to remember the proper order of operation. Here is an example.
Example 7.5
Solve [latex]F=\frac{9}{5}C+32[/latex] for C.
Answer: First, subtract 32 from both sides to get
[latex]F-32=\frac{9}{5}C[/latex]
Then, isolate C by multiplying both sides by 5 and dividing both sides by 9
[latex]\frac{5}{9}*(F-32)=\frac{9}{5}C*\frac{5}{9}[/latex]
Canceling the 5 and 9 on the right gives
[latex]C=\frac{5}{9}(F-32)[/latex]
