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Solving Literal Equations

A literal equation is an equation with more than one variable. Formulas can be considered literal equations. To solve a literal equation, use inverse operations and the Properties of Equality to isolate the variable of interest.
Exercise

Solve Ax+By=CAx+By=C for x.x.

Solution
Notice that the equation above has many different variables. However, xx is the variable of interest. To isolate x,x, we need to move ByBy and AA to the right-hand side of the equation. Recall that only like terms can be combined.
Ax+By=CAx+By=C
Ax+ByBy=CByAx+By-By=C-By
Ax=CByAx=C-By
AxA=CByA\dfrac{Ax}{A}=\dfrac{C-By}{A}
x=CByAx = \dfrac{C-By}{A}
Solved for x,x, the equation is x=CByA.x = \dfrac{C-By}{A}.
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Concept

Rearranging Formulas

Sometimes, rearranging a formula so that a different variable is highlighted, can make solving a problem easier. The same approach as above can be followed to isolate a variable of interest.
Exercise

To convert temperatures in Fahrenheit, F,F, to temperatures in Celsius, C,C, the following formula can be used. F=95C+32 F=\frac{9}{5}C+32 Solve the formula for C.C. Then use the resultant formula to convert 9898 ^\circ Fahrenheit to Celsius. Round to the nearest degree.

Solution
In the given formula, CC is the variable of interest. To isolate C,C, we can move 3232 and 95\frac{9}{5} to the left-hand side using inverse operations.
F=95C+32F=\dfrac{9}{5}C+32
F32=95CF-32=\dfrac{9}{5}C
F32=9C5F-32=\dfrac{9C}{5}
5(F32)=9C5(F-32) = 9C
5(F32)9=C\dfrac{5(F-32)}{9}=C
C=5(F32)9C=\dfrac{5(F-32)}{9}
Thus, the formula we can use to convert degrees Fahrenheit to Celsius is C=5(F32)9.C=\frac{5(F-32)}{9}. Let's use this to find the equivalent of 9898 ^\circ Fahrenheit in Celsius.^\circ \text{Celsius}. To begin, we'll substitute F=98.F=98.
C=5(F32)9C=\dfrac{5(F-32)}{9}
C=5(9832)9C=\dfrac{5({\color{#0000FF}{98}}-32)}{9}
C=5669C=\dfrac{5 \cdot 66}{9}
C=3309C=\dfrac{330}{9}
C=36.66C=36.66\ldots
C=37C=37
Thus, 9898 ^\circ Fahrenheit is approximately equivalent to 3737 ^\circ Celsius.
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