Solve Equations with Decimals

Determine whether a decimal is a solution of an equation

Solving equations with decimals is important in our everyday lives because money is usually written with decimals. When applications involve money — such as shopping for yourself, making your family’s budget, or planning for the future of your business — you’ll be solving equations with decimals.

Now that we’ve worked with decimals, we are ready to find solutions to equations involving decimals. The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number, an integer, a fraction, or a decimal.

Example. Determine whether each of the following is a solution of $x - 0.7 = 1.5$: (a) $x = 1$ (b) $x = -0.8$ (c) $x = 2.2$.

(a) Substitute $1$ for $x$: $1 - 0.7 \stackrel{?}{=} 1.5$. Subtract: $0.3 \neq 1.5$. Since $x = 1$ does not result in a true equation, $1$ is not a solution to the equation.

(b) Substitute $-0.8$ for $x$: $-0.8 - 0.7 \stackrel{?}{=} 1.5$. Subtract: $-1.5 \neq 1.5$. Since $x = -0.8$ does not result in a true equation, $-0.8$ is not a solution to the equation.

(c) Substitute $2.2$ for $x$: $2.2 - 0.7 \stackrel{?}{=} 1.5$. Subtract: $1.5 = 1.5\ \checkmark$. Since $x = 2.2$ results in a true equation, $2.2$ is a solution to the equation.

Solve equations with decimals

In previous chapters, we solved equations using the Properties of Equality. We will use these same properties to solve equations with decimals.

When you add, subtract, multiply, or divide the same quantity from both sides of an equation, you still have equality.

Example. Solve: $y + 2.3 = -4.7$.

We will use the Subtraction Property of Equality to isolate the variable. Subtract $2.3$ from each side, to undo the addition: $y + 2.3 - 2.3 = -4.7 - 2.3$. Simplify: $y = -7$.

Check: substitute $y = -7$: $-7 + 2.3 \stackrel{?}{=} -4.7$. Simplify: $-4.7 = -4.7\ \checkmark$. Since $y = -7$ makes $y + 2.3 = -4.7$ a true statement, we know we have found a solution to this equation.

Example. Solve: $a - 4.75 = -1.39$.

We will use the Addition Property of Equality. Add $4.75$ to each side, to undo the subtraction: $a - 4.75 + 4.75 = -1.39 + 4.75$. Simplify: $a = 3.36$.

Check: substitute $a = 3.36$: $3.36 - 4.75 \stackrel{?}{=} -1.39$. Simplify: $-1.39 = -1.39\ \checkmark$. Since the result is a true statement, $a = 3.36$ is a solution to the equation.

Example. Solve: $-4.8 = 0.8n$.

We will use the Division Property of Equality. We must divide both sides by $0.8$ to isolate $n$: $\tfrac{-4.8}{0.8} = \tfrac{0.8n}{0.8}$. Simplify: $-6 = n$.

Check: substitute $n = -6$: $-4.8 \stackrel{?}{=} 0.8(-6)$. Simplify: $-4.8 = -4.8\ \checkmark$. Since $n = -6$ makes $-4.8 = 0.8n$ a true statement, we know we have a solution.

Example. Solve: $\tfrac{p}{-1.8} = -6.5$.

We will use the Multiplication Property of Equality. Here, $p$ is divided by $-1.8$. We must multiply by $-1.8$ to isolate $p$: $-1.8\left(\tfrac{p}{-1.8}\right) = -1.8(-6.5)$. Multiply: $p = 11.7$.

Check: substitute $p = 11.7$: $\tfrac{11.7}{-1.8} \stackrel{?}{=} -6.5$. Simplify: $-6.5 = -6.5\ \checkmark$. A solution to $\tfrac{p}{-1.8} = -6.5$ is $p = 11.7$.

Translate to an equation and solve

Now that we have solved equations with decimals, we are ready to translate word sentences to equations and solve. Remember to look for words and phrases that indicate the operations to use.

Example. Translate and solve: The difference of $n$ and $4.3$ is $2.1$.

Translate: $n - 4.3 = 2.1$. Add $4.3$ to both sides of the equation: $n - 4.3 + 4.3 = 2.1 + 4.3$. Simplify: $n = 6.4$.

Check: is the difference of $n$ and $4.3$ equal to $2.1$? Let $n = 6.4$: is the difference of $6.4$ and $4.3$ equal to $2.1$? Translate: $6.4 - 4.3 \stackrel{?}{=} 2.1$. Simplify: $2.1 = 2.1\ \checkmark$.

Example. Translate and solve: The product of $-3.1$ and $x$ is $5.27$.

Translate: $-3.1x = 5.27$. Divide both sides by $-3.1$: $\tfrac{-3.1x}{-3.1} = \tfrac{5.27}{-3.1}$. Simplify: $x = -1.7$.

Check: is the product of $-3.1$ and $x$ equal to $5.27$? Let $x = -1.7$: is the product of $-3.1$ and $-1.7$ equal to $5.27$? Translate: $-3.1(-1.7) \stackrel{?}{=} 5.27$. Simplify: $5.27 = 5.27\ \checkmark$.

Example. Translate and solve: The quotient of $p$ and $-2.4$ is $6.5$.

Translate: $\tfrac{p}{-2.4} = 6.5$. Multiply both sides by $-2.4$: $-2.4\left(\tfrac{p}{-2.4}\right) = -2.4(6.5)$. Simplify: $p = -15.6$.

Check: is the quotient of $p$ and $-2.4$ equal to $6.5$? Let $p = -15.6$: is the quotient of $-15.6$ and $-2.4$ equal to $6.5$? Translate: $\tfrac{-15.6}{-2.4} \stackrel{?}{=} 6.5$. Simplify: $6.5 = 6.5\ \checkmark$.

Example. Translate and solve: The sum of $n$ and $2.9$ is $1.7$.

Translate: $n + 2.9 = 1.7$. Subtract $2.9$ from each side: $n + 2.9 - 2.9 = 1.7 - 2.9$. Simplify: $n = -1.2$.

Check: is the sum $n$ and $2.9$ equal to $1.7$? Let $n = -1.2$: is the sum $-1.2$ and $2.9$ equal to $1.7$? Translate: $-1.2 + 2.9 \stackrel{?}{=} 1.7$. Simplify: $1.7 = 1.7\ \checkmark$.

Key terms

solution of an equation — a value that, when substituted for the variable, makes the equation a true statement. Properties of Equality — the Subtraction, Addition, Division, and Multiplication properties: applying the same operation to both sides of a true equation keeps it true, which lets us isolate a variable and solve for it.

Practice

Determine whether a decimal is a solution of an equation

Solve equations with decimals

Translate to an equation and solve

This section is adapted from Prealgebra 2e, Section 5.4: Solve Equations with Decimals by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the Properties of Equality table as a callout and the step-by-step solution/check layouts as prose with typeset math; omitted the Be Prepared quiz and Media links; converted the practice problems (“Try Its”) into interactive exercises with instant feedback; and adapted selected end-of-section exercises into the interactive Practice block, expanding multipart items into one exercise per part.