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5.4 Solve equations with decimals

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By the end of this section, you will be able to:
  • Determine whether a decimal is a solution of an equation
  • Solve equations with decimals
  • Translate to an equation and solve

Before you get started, take this readiness quiz.

  1. Evaluate x + 2 3 when x = − 1 4 .
    If you missed this problem, review Add and Subtract Fractions with Different Denominators .
  2. Evaluate 15 − y when y = −5 .
    If you missed this problem, review Subtract Integers .
  3. Solve n − 7 = 42 .
    If you missed this problem, review Solve Equations with Fractions .

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 a number is a solution to an equation.

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

Determine whether each of the following is a solution of x − 0.7 = 1.5 :

ⓐ x = 1 ⓑ x = −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 = −0.8 does not result in a true equation, −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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Determine whether each value is a solution of the given equation.

x − 0.6 = 1.3 : ⓐ x = 0.7 ⓑ x = 1.9 ⓒ x = −0.7

  1. ⓐ no
  2. ⓑ yes
  3. ⓒ no

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Determine whether each value is a solution of the given equation.

y − 0.4 = 1.7 : ⓐ y = 2.1 ⓑ y = 1.3 ⓒ −1.3

  1. ⓐ yes
  2. ⓑ no
  3. ⓒ no

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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.

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 ≠ 0
If a = b , then a c = b c 		The Multiplication Property of Equality
For any numbers a , b , and c ,
If a = b , then a c = b c

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

Solve: y + 2.3 = −4.7 .

Solution

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

.
. .
Simplify. .
Check: .
. .
Simplify. .

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

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Solve: y + 2.7 = −5.3 .

y = −8

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Solve: y + 3.6 = −4.8 .

y = −8.4

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Solve: a − 4.75 = −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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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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