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Solving Equations with Decimals

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Solving Equations with Decimals

In previous tutorials, 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 \(a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c,\)

 

If \(a=b,\) then \(a-c=b-c.\)

Addition Property of Equality

 

For any numbers \(a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c,\)

 

If \(a=b,\) then \(a+c=b+c.\)

The Division Property of Equality

 

For any numbers \(a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c\ne 0\)

 

If \(a=b,\) then \(\frac{a}{c}=\frac{b}{c}\)

The Multiplication Property of Equality

 

For any numbers \(a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c,\)

 

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

Solve: \(y+2.3=-4.7.\)

Solution

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

 Solving Equations with Decimals
Solving Equations with DecimalsSolving Equations with Decimals
Simplify.Solving Equations with Decimals
Check:Solving Equations with Decimals 
Solving Equations with DecimalsSolving Equations with Decimals 
Simplify.Solving Equations with Decimals 

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

Optional Video: Solve a One Step Equation With Decimals by Adding and Subtracting

Example

Solve: \(a-4.75=-1.39.\)

Solution

We will use the Addition Property of Equality.

 Solving Equations with Decimals
Add 4.75 to each side, to undo the subtraction.Solving Equations with Decimals
Simplify.Solving Equations with Decimals
Check:Solving Equations with Decimals 
Solving Equations with DecimalsSolving Equations with Decimals 
 Solving Equations with Decimals 

Since the result is a true statement, \(a=3.36\) is a solution to the equation.

Optional Video: Solve a One Step Equation With Decimals by Multiplying

Example

Solve: \(-4.8=0.8n.\)

Solution

We will use the Division Property of Equality.

Use the Properties of Equality to find a value for \(n.\)

 Solving Equations with Decimals
We must divide both sides by 0.8 to isolate n.Solving Equations with Decimals
Simplify.Solving Equations with Decimals
Check:Solving Equations with Decimals 
Solving Equations with DecimalsSolving Equations with Decimals 
 Solving Equations with Decimals 

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

Optional Video: Solve a One Step Equation With Decimals by Dividing

Example

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

Solution

We will use the Multiplication Property of Equality.

 Solving Equations with Decimals
Here, p is divided by −1.8. We must multiply by −1.8 to isolate pSolving Equations with Decimals
Multiply.Solving Equations with Decimals
Check:Solving Equations with Decimals 
Solving Equations with DecimalsSolving Equations with Decimals 
 Solving Equations with Decimals 

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

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