5.4 Solve Equations with Decimals

The topics covered in this section are:

Determine whether a decimal is a solution of an equation
Solve equations with decimals
Translate to an equation and solve

5.4.1 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.

HOW TO: Determine whether a number is a solution to an equation.

Substitute the number for the variable in the equation.
Simplify the expressions on both sides of the equation.
Determine whether the resulting equation is true.
- If so, the number is a solution.
- If not, the number is not a solution.

Example 1

Determine whether each of the following is a solution of $x-0.7=1.5$.

$x=1$
$x=-0.8$
$x=2.2$

Solution

Part 1.
	$x-0.7=1.5$
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.

Part 2.
	$x-0.7=1.5$
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.

Part 3.
	$x-0.7=1.5$
Substitute $-0.8$ for $x$.	$2.2-0.7 \stackrel{?}{=} 1.5$
Subtract.	$1.5=1.5$✓

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

5.4.2 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.

PROPERTIES OF EQUALITY

Subtraction Property of Equality For any numbers $a,b$, and $c$, If $a=b$, then $a-c=b-c$.	Addition Property of Equality For any numbers $a,b$, and $c$, If $a=b$, then $a+c=b+c$.
The Division Property of Equality For any numbers $a,b$, and $c$, and $c \neq 0$ If $a=b$, then $\frac{a}{c} = \frac{b}{c}$.	The Multiplication Property of Equality For any numbers $a,b$, and $c$, If $a=b$, then $ac=bc$.

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

Example 2

Solve: $y+2.3=-4.7$.

Solution

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

	$y+2.3=-4.7$
Subtract $2.3$ from each side, to undo the addition.	$y+2.3-2.3=-4.7-2.3$
Simplify.	$y=-7$

Check.	$y+2.3=-4.7$
Substitute $y=-7$.	$-7+2.3 \stackrel{?}{=} -4.7$
Simplify.	$-4.7=-4.7$✓

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

Example 3

Solve: $a-4.75=-1.39$.

Solution

We will use the Addition Property of Equality.

	$a-4.75=-1.39$
Add $4.75$ each side, to undo the subtraction.	$a-4.75+4.75=-1.39+4.75$
Simplify.	$a=3.36$

Check.	$a-4.75=-1.39$
Substitute $a=3.36$.	$3.36-4.75 \stackrel{?}{=} -1.39$
Simplify.	$-1.39=-1.36$✓

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

Example 4

Solve: $-4.8=0.8n$.

Solution

We will use the Division Property of Equality.

Use the Properties of Equality to find a value for $n$.

	$-4.8=0.8n$
We must divide both sides by $0.8$ to isolate $n$.	$\frac{-4.8}{0.8} = \frac{0.8n}{0.8}$
Simplify.	$-6=n$

Check.	$-4.8=0.8n$
Substitute $n=-6$.	$-4.8 \stackrel{?}{=} 0.8(-6)$
	$-4.8=-4.8$✓

Since $n=-6$ makes $-4.8=0.8n$ a true statement, we know we have a solution.

Example 5

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

Solution

We will use the Multiplication Property of Equality.

	$\frac{p}{-1.8} = -6.5$
Here, $p$ is divided by $-1.8$. We must multiply by $-1.8$ to isolate $p$.	$-1.8 ( \frac{p}{-1.8} ) = -1.8(-6.5)$
Simplify.	$p=11.7$

Check.	$\frac{p}{-1.8} = -6.5$
Substitute $p=11.7$.	$\frac{11.7}{-1.8} \stackrel{?}{=} -6.5$
	$-6.5=-6.5$✓

A solution to $\frac{p}{-1.8} = -6.5$ is $p=11.7$.

5.4.3 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 6

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

Solution

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.	$2.1=2.1$✓

Example 7

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

Solution

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.	$5.27=5.27$✓

Example 8

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

Solution

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.	$\frac{-15.6}{-2.4} \stackrel{?}{=} 6.5$
Simplify.	$6.5=6.5$✓

Example 9

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

Solution

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.	$1.7=1.7$✓