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1. Equations & Inequalities
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Algebra 1

1.05 Literal equations

Lesson
Practice

Literal equations

Literal equation

A formula or equation that consists primarily of variables.

Formula

A type of literal equation that describes a relationship between real-world quantities.

Example:

A = l\cdot w

where A is area,

l is length, and

w is width.

There are many different formulas in science, mathematics, business, and other subjects that allow us to measure quantities such as area, volume, speed, etc. We can use the properties of equality to isolate any variable in a literal equation or formula.

The same variable might be used to represent different quantities across 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. To avoid any confusion, formulas will always state what the variables represent.

A variable can also act as a placeholder for other expressions when a formula applies in a variety of different situations. For example, the formula for the volume of both a cone and a pyramid is

V=\dfrac{1}{3}Bh

where B represents the area of the two-dimensional shape at the base and h is the vertical height from the base. To find the volume of each specific figure, we replace the B with the area formulas for the base shapes.

A square-based pyramid with height h. The area A is equal to s squared.
Square-based pyramid = \dfrac{1}{3}s^{2}h
A right cone with radius r and height h. The area A is equal to pi r squared.
Cone = \dfrac{1}{3}\pi r^{2}h

Exploration

Consider the formula for distance:

\displaystyle d=rt
\bm{d}
distance
\bm{r}
rate
\bm{t}
time

Use the distance formula to solve each of the following:

  • The speed at which a person travels if they drive 100 \text{ mi} in 75 \text{ min}
  • The distance traveled when walking 5 \text{ km} in 30 \text{ min}
  • The distance traveled by a car driving 65 \text{ mph} for 5 \text{ hr}
  • The time it will take to travel 1\,000 \text{ km} if you walk 80 \text{ m/min}
  • How fast a plane is traveling if it can fly 2\,789 \text{ mi} in 6 \text{ hr}
  • How long it will take to walk to the store 2 \text{ mi} away if you walk 176 \text{ ft/min}

  1. Which ones took the most effort to solve?

  2. How could we reduce the effort when making repeated calculations for the same variable?

Using the division property of equality, we can rearrange the equation relating distance, rate, and time to be r=\dfrac{d}{t} or t=\dfrac{d}{r}.

By rearranging a formula for a variable of interest, we can reduce the number of repeated calculations needed, depending on which variable is unknown.

Examples

Example 1

Ohm's law states:

V=IR

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

Write the formula for current.

Worked Solution
Create a strategy

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

Apply the idea
\displaystyle V\displaystyle =\displaystyle IRGiven equation
\displaystyle \dfrac{V}{R}\displaystyle =\displaystyle IDivision property of equality
\displaystyle I\displaystyle =\displaystyle \dfrac{V}{R}Symmetric property of equality

Example 2

Solve for x in the following equation:

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

Worked Solution
Create a strategy

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.

Apply the idea
\displaystyle y\displaystyle =\displaystyle 5 \left(1+\dfrac{x}{k}\right)Given equation
\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
Reflect and check

Remember, 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 3

To find the sum, S, of the interior angles of any polygon with n sides, the following equation can be used:

S=180\left(n-2\right)

a

A polygon's angles sum to 900 \degree. Use the sum of interior angles formula to determine its number of sides.

Worked Solution
Create a strategy

Since S is the sum of the angles, replace the S in S=180\left(n-2\right) with 900 and solve.

Apply the idea

Since the angles sum to 900 \degree, the S in the formula will be repleaced with 900 and we will solve for n.

\displaystyle 900\displaystyle =\displaystyle 180\left(n-2\right)Replace S with 900
\displaystyle \frac{900}{180}\displaystyle =\displaystyle \frac{180\left(n-2\right)}{180}Divide both sides by 180
\displaystyle 5\displaystyle =\displaystyle n-2
\displaystyle 5+2\displaystyle =\displaystyle n-2+2Add 2 to each side
\displaystyle 7\displaystyle =\displaystyle nSimplify

A polygon whose angles sum to 900 \degree has 7 sides.

Reflect and check

An alternative but equivalent way to solve for n once S is replaced with 900 is to distribute first then solve.

\displaystyle 900\displaystyle =\displaystyle 180\left(n-2\right)Replace S with 900
\displaystyle 900\displaystyle =\displaystyle 180n-360Distribute
\displaystyle 900+360\displaystyle =\displaystyle 180n-360+360Add 360 to both sides
\displaystyle 1260\displaystyle =\displaystyle 180n
\displaystyle \frac{1260}{180}\displaystyle =\displaystyle \frac{180n}{180}Divide by 180
\displaystyle 7\displaystyle =\displaystyle n
b

Write an equation to solve for n using the properties of equality.

Worked Solution
Create a strategy

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

Apply the idea
\displaystyle S\displaystyle =\displaystyle 180\left(n-2\right)Given equation
\displaystyle S\displaystyle =\displaystyle 180n - 360Distributive property
\displaystyle S+360\displaystyle =\displaystyle 180nAddition property of equality
\displaystyle \frac{S+360}{180}\displaystyle =\displaystyle nDivision property of equality
\displaystyle n\displaystyle =\displaystyle \frac{S}{180}+2Reflexive property

The equation S=\left(n-2\right) solved for n is n=\dfrac{S}{180}+2.

Reflect and check

Equations can be represented multiple ways, and the simplified equation n=\dfrac{S}{180}+2 is also equivalent to the equation found in the second to last step, \dfrac{S+360}{180}=n.

After applying the reflexive property, n=\dfrac{S+360}{180} represents an equivalent and correct way to solve for n.

c

Use the equation from part (b) to find the number of sides of a polygon whose angles sum to 1440 \degree.

Worked Solution
Create a strategy

Replace S with 1440 and evaluate.

Apply the idea
\displaystyle n\displaystyle =\displaystyle \frac{S}{180}+2Original equation
\displaystyle n\displaystyle =\displaystyle \frac{1440}{180}+2Substitute S=1440
\displaystyle n\displaystyle =\displaystyle 8+2Divide
\displaystyle n\displaystyle =\displaystyle 10Simplify

A polygon whose angles sum to 1440 \degree has 10 sides.

Reflect and check

Using the original equation equation S=180\left(n-2\right) can also be used to find the number of sides, and will give the same value as n=\dfrac{S}{180}+2.

\displaystyle S\displaystyle =\displaystyle 180\left(n-2\right)Original equation
\displaystyle 1440\displaystyle =\displaystyle 180\left(n-2\right)Replace S with 1440
\displaystyle 1440\displaystyle =\displaystyle 180n-360Distribute
\displaystyle 1800\displaystyle =\displaystyle 180nAdd 360 to both sides
\displaystyle \frac{1800}{180}\displaystyle =\displaystyle \frac{180n}{180}Divide both sides by 180
\displaystyle 10\displaystyle =\displaystyle nSimplify

Using the original equation helps confirm the correct steps were taken to isolate the needed variable.

Idea summary

In the same way we solve one-variable equations, we can use inverse operations and the properties of equality to isolate a variable in a literal equation.

Outcomes

A.EI.1

The student will represent, solve, explain, and interpret the solution to multistep linear equations and inequalities in one variable and literal equations for a specified variable.

A.EI.1d

Rearrange a formula or literal equation to solve for a specified variable by applying the properties of equality.

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