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One-Variable Equations

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Algebra 1
One-Variable Equations
Solving Literal Equations

Solving Literal Equations

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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
LHS−By=RHS−By\text{LHS}-By=\text{RHS}-By
Ax+By−By=C−ByAx+By-By=C-By
Subtract terms
Ax=C−ByAx=C-By
LHS/A=RHS/A\left.\text{LHS}\middle/A\right.=\left.\text{RHS}\middle/A\right.
AxA=C−ByA\dfrac{Ax}{A}=\dfrac{C-By}{A}
Simplify quotient
x=C−ByAx = \dfrac{C-By}{A}
Solved for x,x, the equation is x=C−ByA.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 98∘98 ^\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
LHS−32=RHS−32\text{LHS}-32=\text{RHS}-32
F−32=95CF-32=\dfrac{9}{5}C
ac⋅b=a⋅bc\dfrac{a}{c}\cdot b = \dfrac{a\cdot b}{c}
F−32=9C5F-32=\dfrac{9C}{5}
LHS⋅5=RHS⋅5\text{LHS} \cdot 5=\text{RHS}\cdot 5
5(F−32)=9C5(F-32) = 9C
LHS/9=RHS/9\left.\text{LHS}\middle/9\right.=\left.\text{RHS}\middle/9\right.
5(F−32)9=C\dfrac{5(F-32)}{9}=C
Rearrange equation
C=5(F−32)9C=\dfrac{5(F-32)}{9}
Thus, the formula we can use to convert degrees Fahrenheit to Celsius is C=5(F−32)9.C=\frac{5(F-32)}{9}. Let's use this to find the equivalent of 98∘98 ^\circ Fahrenheit in ∘Celsius.^\circ \text{Celsius}. To begin, we'll substitute F=98.F=98.
C=5(F−32)9C=\dfrac{5(F-32)}{9}
F=98F={\color{#0000FF}{98}}
C=5(98−32)9C=\dfrac{5({\color{#0000FF}{98}}-32)}{9}
Subtract term
C=5⋅669C=\dfrac{5 \cdot 66}{9}
Multiply
C=3309C=\dfrac{330}{9}
Use a calculator
C=36.66…C=36.66\ldots
Round to nearest integer
C=37C=37
Thus, 98∘98 ^\circ Fahrenheit is approximately equivalent to 37∘37 ^\circ Celsius.
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Exercises

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