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Solving Literal Equations

One-Variable 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.

Solve the literal equation

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Exercise

Solve Ax+By=C for x.

Show Solution

Solution

Notice that the equation above has many different variables. However, x is the variable of interest. To isolate x, we need to move By and A to the right-hand side of the equation. Recall that only like terms can be combined.

Ax+By=C

LHS−By=RHS−By

Ax+By−By=C−By

Subtract terms

Ax=C−By

$LHS / A = RHS / A$

\frac{A x}{A} = \frac{C - B y}{A}

Simplify quotient

x = \frac{C - B y}{A}

Solved for x, the equation is

x = \frac{C - B y}{A} .

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.

Solve the formula

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Exercise

To convert temperatures in Fahrenheit, F, to temperatures in Celsius, C, the following formula can be used.

Solve the formula for C. Then use the resultant formula to convert

9 8^{\circ}

Fahrenheit to Celsius. Round to the nearest degree.

Show Solution

Solution

In the given formula, C is the variable of interest. To isolate C, we can move 32 and

\frac{9}{5}

to the left-hand side using inverse operations.

F = \frac{9}{5} C + 32

LHS−32=RHS−32

F - 32 = \frac{9}{5} C

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

F - 32 = \frac{9 C}{5}

LHS⋅5=RHS⋅5

5(F−32)=9C

$LHS / 9 = RHS / 9$

\frac{5 ( F - 3 2 )}{9} = C

Rearrange equation

C = \frac{5 ( F - 3 2 )}{9}

Thus, the formula we can use to convert degrees Fahrenheit to Celsius is

C = \frac{5 ( F - 3 2 )}{9} .

Let's use this to find the equivalent of

9 8^{\circ}

Fahrenheit in

^{\circ} Celsius .

To begin, we'll substitute F=98.

C = \frac{5 ( F - 3 2 )}{9}

F=98

C = \frac{5 ( 9 8 - 3 2 )}{9}

Subtract term

C = \frac{5 \cdot 6 6}{9}

Multiply

C = \frac{3 3 0}{9}

Use a calculator

C = 36.66 \dots

Round to nearest integer

C=37

Thus,

9 8^{\circ}

Fahrenheit is approximately equivalent to

3 7^{\circ}

Celsius.

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