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.



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.

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.


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.

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 \(n.\)


We must divide both sides by 0.8 to isolate n.
Simplify.
Check:

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.


Here, p is divided by −1.8. We must multiply by −1.8 to isolate p
Multiply.
Check:

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

This lesson is part of:

Introducing Decimals

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