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1. Equations & Inequalities in One Variable
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United States of AmericaTennessee
Integrated Math 1

1.04 Literal equations

Lesson
Practice
Lesson

Concept summary

Some equations involve more than one variable. These are usually still referred to as just equations, but sometimes they are given other names.

Literal equation

An equation that involves two or more variables

Example:

x = 5y+3z

We can use the properties of equality to isolate any variable in a literal equation.

Formula

A type of literal equation that describes a relationship between certain quantites

Example:

A = l\cdot w

where A is area,

l is length, and

w is width.

The same variable might be used to represent different quantities acorss different formulas. For example, in the formula for the area of a rectangle w is used to represent the width of the rectangle, however in another context w might be used to represent a weight or other value.

Worked examples

Example 1

Solve for x in the following equation:

y = 5 \left(1+\dfrac{x}{k}\right)

Approach

We need to rearrange the equation to isolate x. We can use the properties of equality and inverse operations to solve literal equations for a variable, just as we would for linear equations.

Solution

\displaystyle y\displaystyle =\displaystyle 5 \left(1+\dfrac{x}{k}\right)
\displaystyle \dfrac{y}{5}\displaystyle =\displaystyle 1+\dfrac{x}{k}Division property of equality
\displaystyle \dfrac{y}{5} -1\displaystyle =\displaystyle \dfrac{x}{k}Subtraction property of equality
\displaystyle k\left(\dfrac{y}{5} -1\right)\displaystyle =\displaystyle xMultiplication property of equality
\displaystyle x\displaystyle =\displaystyle k\left(\dfrac{y}{5} -1\right)Symmetric property of equality

Reflection

When rearranging an equation, we reverse the operations acting on the variable we want to isolate in the reverse order of operations. Whatever is done to one side of the equation, must be done to the other to keep the equation balanced.

Example 2

Given the formula for Ohm's law:

V=IR

where V is voltage, I is current and R is resistance.

Write the formula for current.

Approach

The formula for current is Ohm's law with I isolated. We use inverse operations and properties of equality to get the solution.

Solution

\displaystyle V\displaystyle =\displaystyle IR
\displaystyle \dfrac{V}{R}\displaystyle =\displaystyle IDivision property of equality
\displaystyle I\displaystyle =\displaystyle \dfrac{V}{R}Symmetric property of equality

Example 3

The perimeter of a semicircle is given by the formula:

P = 2r+2 \pi r

Write an equation to solve for r.

Approach

We need to rearrange the equation to isolate r. We can use the properties of equality and inverse operations to get the solution.

Solution

\displaystyle P\displaystyle =\displaystyle 2r+2 \pi r
\displaystyle P\displaystyle =\displaystyle r \left(2+2 \pi\right)Distributive property
\displaystyle \dfrac{P}{\left(2+2 \pi\right)}\displaystyle =\displaystyle rDivision property of equality
\displaystyle r\displaystyle =\displaystyle \dfrac{P}{\left(2+2 \pi\right)}Symmetric property of equality

Reflection

Notice the variable we are trying to solve for was in more than one term, if these terms cannot be combined we may be required to factor part of the expression to isolate the variable.

Outcomes

M1.N.Q.A.1

Use units as a way to understand real-world problems.*

M1.N.Q.A.1.B

Use appropriate quantities in formulas, converting units as necessary.*

M1.A.CED.A.4

Rearrange formulas to isolate a quantity of interest using algebraic reasoning.

M1.A.REI.A.1

Understand solving equations as a process of reasoning and explain the reasoning. Construct a viable argument to justify a solution method.

M1.MP1

Make sense of problems and persevere in solving them.

M1.MP3

Construct viable arguments and critique the reasoning of others.

M1.MP4

Model with mathematics.

M1.MP6

Attend to precision.

M1.MP7

Look for and make use of structure.

M1.MP8

Look for and express regularity in repeated reasoning.

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