Mathematics » Solving Linear Equations I » Solve Equations Using the Division and Multiplication Properties of Equality

Solving Equations Using the Division and Multiplication Properties of Equality

Solving Equations Using the Division and Multiplication Properties of Equality

We introduced the Multiplication and Division Properties of Equality in Mathematics 103 and Mathematics 104. We modeled how these properties worked using envelopes and counters and then applied them to solving equations (See Mathematics 103). We restate them again here as we prepare to use these properties again.

Definition: Division and Multiplication Properties of Equality

Division Property of Equality: For all real numbers \(a,b,c,\) and \(c\ne 0,\) if \(a=b,\) then \(\frac{a}{c}=\frac{b}{c}.\)

Multiplication Property of Equality: For all real numbers \(a,b,c,\) if \(a=b,\) then \(ac=bc.\)

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

Let’s review how these properties of equality can be applied in order to solve equations. Remember, the goal is to ‘undo’ the operation on the variable. In the example below the variable is multiplied by \(4,\) so we will divide both sides by \(4\) to ‘undo’ the multiplication.

Optional Video: Solving One Step Equation by Mult/Div. Integers (Var on Left)

Example

Solve: \(4x=-28.\)

Solution

We use the Division Property of Equality to divide both sides by \(4.\)

 Solving Equations Using the Division and Multiplication Properties of Equality
Divide both sides by 4 to undo the multiplication.Solving Equations Using the Division and Multiplication Properties of Equality
Simplify.Solving Equations Using the Division and Multiplication Properties of Equality
Check your answer. Let \(x=-7\). 
Solving Equations Using the Division and Multiplication Properties of Equality 
Solving Equations Using the Division and Multiplication Properties of Equality 
Solving Equations Using the Division and Multiplication Properties of Equality 

Since this is a true statement, \(x=-7\) is a solution to \(4x=-28.\)

In the previous example, to ‘undo’ multiplication, we divided. How do you think we ‘undo’ division?

Example

Solve: \(\frac{\phantom{\rule{0.4em}{0ex}}a}{-7}=-42.\)

Solution

Here \(a\) is divided by \(-7.\) We can multiply both sides by \(-7\) to isolate \(a.\)

 Solving Equations Using the Division and Multiplication Properties of Equality
Multiply both sides by \(-7\).Solving Equations Using the Division and Multiplication Properties of Equality

 

Solving Equations Using the Division and Multiplication Properties of Equality

Simplify.Solving Equations Using the Division and Multiplication Properties of Equality
Check your answer. Let \(a=294\). 
Solving Equations Using the Division and Multiplication Properties of Equality 
Solving Equations Using the Division and Multiplication Properties of Equality 
Solving Equations Using the Division and Multiplication Properties of Equality 

Example

Solve: \(-r=2.\)

Solution

Remember \(-r\) is equivalent to \(-1r.\)

 Solving Equations Using the Division and Multiplication Properties of Equality
Rewrite \(-r\) as \(-1r\).Solving Equations Using the Division and Multiplication Properties of Equality
Divide both sides by \(-1\).Solving Equations Using the Division and Multiplication Properties of Equality
 Solving Equations Using the Division and Multiplication Properties of Equality
Check.Solving Equations Using the Division and Multiplication Properties of Equality 
Substitute \(r=-2\)Solving Equations Using the Division and Multiplication Properties of Equality 
Simplify.Solving Equations Using the Division and Multiplication Properties of Equality 

In Mathematics 104, we saw that there are two other ways to solve \(-r=2.\)

We could multiply both sides by \(-1.\)

We could take the opposite of both sides.

Example

Solve: \(\frac{2}{3}\phantom{\rule{0.1em}{0ex}}x=18.\)

Solution

Since the product of a number and its reciprocal is \(1,\) our strategy will be to isolate \(x\) by multiplying by the reciprocal of \(\frac{2}{3}.\)

 Solving Equations Using the Division and Multiplication Properties of Equality
Multiply by the reciprocal of \(\frac{2}{3}\).Solving Equations Using the Division and Multiplication Properties of Equality
Reciprocals multiply to one.Solving Equations Using the Division and Multiplication Properties of Equality
Multiply.Solving Equations Using the Division and Multiplication Properties of Equality
Check your answer. Let \(x=27\) 
Solving Equations Using the Division and Multiplication Properties of Equality 
Solving Equations Using the Division and Multiplication Properties of Equality 
Solving Equations Using the Division and Multiplication Properties of Equality 

Notice that we could have divided both sides of the equation \(\frac{2}{3}\phantom{\rule{0.1em}{0ex}}x=18\) by \(\frac{2}{3}\) to isolate \(x.\) While this would work, multiplying by the reciprocal requires fewer steps.

Optional Video: Solving One Step Equation by Mult/Div. Integers (Var on Right)

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