5.4 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. We’ll list these steps here again for easy reference.

Determine whether each of the following is a solution of $x-0.7=1.5\text{:}$

ⓐ $x=1$ ⓑ $x=\mathrm{-0.8}$ ⓒ $x=2.2$

Solution

ⓐ
Subtract.

Since $x=1$ does not result in a true equation, $1$ is not a solution to the equation.

ⓑ
Subtract.

Since $x=\mathrm{-0.8}$ does not result in a true equation, $\mathrm{-0.8}$ is not a solution to the equation.

ⓒ
Subtract.

Since $x=2.2$ results in a true equation, $2.2$ is a solution to the equation.

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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.

Solve: $y+2.3=\mathrm{-4.7}.$

Solution

We will use the Subtraction Property of Equality to isolate the variable.

Simplify.
Check:
Simplify.

Since $y=\mathrm{-7}$ makes $y+2.3=\mathrm{-4.7}$ a true statement, we know we have found a solution to this equation.

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Solve: $a-4.75=\mathrm{-1.39}.$

Solution

We will use the Addition Property of Equality.

Add 4.75 to each side, to undo the subtraction.
Simplify.
Check:

Since the result is a true statement, $a=3.36$ is a solution to the equation.

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