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Study Guides > Prealgebra

Using the Division and Multiplication Properties of Equality for Single-Step Equations

Learning Outcomes

  • Review and use the division and multiplication properties of equality to solve linear equations
  • Use a reciprocal to solve a linear equation that contains fractions
  Let's review the Division and Multiplication Properties of Equality as we prepare to use them to solve single-step equations.

Division Property of Equality

For all real numbers a,b,ca,b,c, and c≠0c\ne 0, if a=ba=b, then ac=bc\frac{a}{c}=\frac{b}{c}.

Multiplication Property of Equality

For all real numbers a,b,ca,b,c, if a=ba=b, then ac=bcac=bc.
  Stated simply, 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 44, so we will divide both sides by 44 to "undo" the multiplication.

example

Solve: 4x=−284x=-28. Solution: To solve this equation, we use the Division Property of Equality to divide both sides by 44.
4x=−284x=-28
Divide both sides by 4 to undo the multiplication. 4x4=−284\frac{4x}{\color{red}4}=\frac{-28}{\color{red}4}
Simplify. x=−7x =-7
Check your answer. 4x=−284x=-28
Let x=−7x=-7. Substitute −7-7 for x. 4(−7)=?−284(\color{red}{-7})\stackrel{\text{?}}{=}-28
 −28=−28-28=-28
Since this is a true statement, x=−7x=-7 is a solution to 4x=−284x=-28.
Now you can try to solve an equation that requires division and includes negative numbers.   In the previous example, to "undo" multiplication, we divided. How do you think we "undo" division? Next, we will show an example that requires us to use multiplication to undo division.

example

Solve: a−7=−42\frac{a}{-7}=-42.

Answer: Solution: Here aa is divided by −7-7. We can multiply both sides by −7-7 to isolate aa.

a−7=−42\frac{a}{-7}=-42
Multiply both sides by −7-7 . −7(a−7)=−7(−42)\color{red}{-7}(\frac{a}{-7})=\color{red}{-7}(-42) −7a−7=294\frac{-7a}{-7}=294
Simplify. a=294a=294
Check your answer. a−7=−42\frac{a}{-7}=-42
Let a=294a=294 . 294−7=?−42\frac{\color{red}{294}}{-7}\stackrel{\text{?}}{=}-42
−42=−42✓-42=-42\quad\checkmark

Now see if you can solve a problem that requires multiplication to undo division. Recall the rules for multiplying two negative numbers - two negatives give a positive when they are multiplied.   As you begin to solve equations that require several steps you may find that you end up with an equation that looks like the one in the next example, with a negative variable.  As a standard practice, it is good to ensure that variables are positive when you are solving equations. The next example will show you how.

example

Solve: −r=2-r=2.

Answer: Solution: Remember −r-r is equivalent to −1r-1r.

−r=2-r=2
Rewrite −r-r as −1r-1r . −1r=2-1r=2
Divide both sides by −1-1 . −1r−1=2−1\frac{-1r}{\color{red}{-1}}=\frac{2}{\color{red}{-1}}
Simplify. r=−2r=-2
Check. −r=2-r=2
Substitute r=−2r=-2 −(−2)=?2-(\color{red}{-2})\stackrel{\text{?}}{=}2
Simplify. 2=2✓2=2\quad\checkmark

Now you can try to solve an equation with a negative variable. In our next example, we are given an equation that contains a variable multiplied by a fraction. We will use a reciprocal to isolate the variable.

example

Solve: 23x=18\frac{2}{3}x=18.

Answer: Solution: Since the product of a number and its reciprocal is 11, our strategy will be to isolate xx by multiplying by the reciprocal of 23\frac{2}{3}.

23x=18\frac{2}{3}x=18
Multiply by the reciprocal of 23\frac{2}{3} . 32⋅23x\frac{\color{red}{3}}{\color{red}{2}}\cdot\frac{2}{3}x
Reciprocals multiply to one. 1x=32⋅1811x=\frac{3}{2}\cdot\frac{18}{1}
Multiply. x=27x=27
Check your answer. 23x=18\frac{2}{3}x=18
Let x=27x=27. 23⋅27=?18\frac{2}{3}\cdot\color{red}{27}\stackrel{\text{?}}{=}18
18=18✓18=18\quad\checkmark

Notice that we could have divided both sides of the equation 23x=18\frac{2}{3}x=18 by 23\frac{2}{3} to isolate xx. While this would work, multiplying by the reciprocal requires fewer steps.   The next video includes examples of using the division and multiplication properties to solve equations with the variable on the right side of the equal sign. https://youtu.be/TB1rkPbF8rA

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