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Solve Equations Using the Division and Multiplication Properties of Equality

Solve equations using the Division and Multiplication Properties of Equality

Solve Equations Using the Division and Multiplication Properties of Equality

We introduced the Multiplication and Division Properties of Equality in Solve Equations Using Integers; The Division Property of Equality and Solve Equations with Fractions. We modeled how these properties worked using envelopes and counters and then applied them to solving equations (See Solve Equations Using Integers; The Division Property of Equality). We restate them again here as we prepare to use these properties again.

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.

Example

Try it.

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

Solution

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


Divide both sides by 4 to undo the multiplication.
Simplify.
Check your answer. Let \(x=-7\).

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

Try it.

Solve: \(\frac{\ a}{-7}=-42.\)

Solution

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


Multiply both sides by \(-7\).
Simplify.
Check your answer. Let \(a=294\).

Example

Try it.

Solve: \(-r=2.\)

Solution

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


Rewrite \(-r\) as \(-1r\).
Divide both sides by \(-1\).

Check.
Substitute \(r=-2\)
Simplify.

In Solve Equations with Fractions, 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.

Condensed — the full section is in OpenStax Prealgebra 2e.

Solve Equations That Need to be Simplified

Many equations start out more complicated than the ones we’ve just solved. First, we need to simplify both sides of the equation as much as possible

Example

Try it.

Solve: \(8x+9x-5x=-3+15.\)

Solution

Start by combining like terms to simplify each side.


Combine like terms.
Divide both sides by 12 to isolate x.
Simplify.
Check your answer. Let \(x=1\)

Example

Try it.

Solve: \(11-20=17y-8y-6y.\)

Solution

Simplify each side by combining like terms.


Simplify each side.
Divide both sides by 3 to isolate y.
Simplify.
Check your answer. Let \(y=-3\)

Notice that the variable ended up on the right side of the equal sign when we solved the equation. You may prefer to take one more step to write the solution with the variable on the left side of the equal sign.

Example

Try it.

Solve: \(-3(n-2)-6=21.\)

Solution

Remember—always simplify each side first.


Distribute.
Simplify.
Divide both sides by -3 to isolate n.
Check your answer. Let \(n=-7\).

Key Concepts

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 \(ac=bc\).

Solve Equations Using the Division and Multiplication Properties of Equality

Solve Equations Using the Division and Multiplication Properties of Equality

In the following exercises, solve each equation for the variable using the Division Property of Equality and check the solution.

Try it.

\(8x=32\)

Try it.

\(7p=63\)

Solution

p = 9

Try it.

\(-5c=55\)

Try it.

\(-9x=-27\)

Solution

x = 3

Try it.

\(-90=6y\)

Try it.

\(-72=12y\)

Solution

y = −6

Try it.

\(-16p=-64\)

Try it.

\(-8m=-56\)

Solution

m = 7

Try it.

\(0.25z=3.25\)

Try it.

\(0.75a=11.25\)

Solution

a = 15

Try it.

\(-3x=0\)

Try it.

\(4x=0\)

Solution

x = 0

In the following exercises, solve each equation for the variable using the Multiplication Property of Equality and check the solution.

Try it.

\(\frac{x}{4}=15\)

Try it.

\(\frac{z}{2}=14\)

Solution

z = 28

Try it.

\(-20=\frac{\ q}{-5}\)

Try it.

\(\frac{\ c}{-3}=-12\)

Solution

c = 36

Try it.

\(\frac{y}{9}=-6\)

Try it.

\(\frac{q}{6}=-8\)

Solution

q = −48

Try it.

\(\frac{m}{-12}=5\)

Try it.

\(-4=\frac{p}{-20}\)

Solution

p = 80

Try it.

\(\frac{2}{3}\ y=18\)

Try it.

\(\frac{3}{5}\ r=15\)

Solution

r = 25

Try it.

\(-\frac{5}{8}\ w=40\)

Try it.

\(24=-\frac{3}{4}\ x\)

Solution

x = −32

Try it.

\(-\frac{2}{5}=\frac{1}{10}\ a\)

Try it.

\(-\frac{1}{3}\ q=-\frac{5}{6}\)

Solution

\(q=\frac{5}{2}\)

Solve Equations That Need to be Simplified

In the following exercises, solve the equation.

Try it.

\(8a+3a-6a=-17+27\)

Try it.

\(6y-3y+12y=-43+28\)

Solution

y = −1

Try it.

\(-9x-9x+2x=50-2\)

Try it.

\(-5m+7m-8m=-6+36\)

Solution

m = −5

Try it.

\(100-16=4p-10p-p\)

Try it.

\(-18-7=5t-9t-6t\)

Solution

\(t=\frac{5}{2}\)

Try it.

\(\frac{7}{8}\ n-\frac{3}{4}\ n=9+2\)

Try it.

\(\frac{5}{12}\ q+\frac{1}{2}\ q=25-3\)

Solution

q = 24

Try it.

\(0.25d+0.10d=6-0.75\)

Try it.

\(0.05p-0.01p=2+0.24\)

Solution

p = 56

Try it.

Frida started to solve the equation \(-3x=36\) by adding \(3\) to both sides. Explain why Frida’s method will result in the correct solution.

Try it.

Emiliano thinks \(x=40\) is the solution to the equation \(\frac{1}{2}\ x=80.\) Explain why he is wrong.

Solution

Answer will vary.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Simplify: \(-7(\frac{1}{-7}).\)
If you missed this problem, review .

Giải đáp

\(1\)
What is the reciprocal of \(-\frac{3}{8}?\)
If you missed this problem, review .

Giải đáp

\(-\frac{8}{3}\)
Evaluate \(9x+2\) when \(x=-3.\)
If you missed this problem, review .

Giải đáp

\(-25\)
Solve: \(4x=-28.\)

Giải đáp

We use the Division Property of Equality to divide both sides by \(4.\)
Divide both sides by 4 to undo the multiplication.
Simplify.
Check your answer. Let \(x=-7\).
Since this is a true statement, \(x=-7\) is a solution to \(4x=-28.\)
Solve: \(3y=-48.\)

Giải đáp

y = −16
Solve: \(4z=-52.\)

Giải đáp

z = −13
Solve: \(\frac{\ a}{-7}=-42.\)

Giải đáp

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

Simplify.
Check your answer. Let \(a=294\).
Solve: \(\frac{\ b}{-6}=-24.\)

Giải đáp

b = 144
Solve: \(\frac{\ c}{-8}=-16.\)

Giải đáp

c = 128
Solve: \(-r=2.\)

Giải đáp

Remember \(-r\) is equivalent to \(-1r.\)
Rewrite \(-r\) as \(-1r\).
Divide both sides by \(-1\).
Check.
Substitute \(r=-2\)
Simplify.
In Solve Equations with Fractions, 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.
Solve: \(-k=8.\)

Giải đáp

k = −8
Solve: \(-g=3.\)

Giải đáp

g = −3
Solve: \(\frac{2}{3}\ x=18.\)

Giải đáp

Since the product of a number and its reciprocal is \(1,\) our strategy will be to isolate \(x\) by multiplying by the reciprocal of \(\frac{2}{3}.\)
Multiply by the reciprocal of \(\frac{2}{3}\).
Reciprocals multiply to one.
Multiply.
Check your answer. Let \(x=27\)
Notice that we could have divided both sides of the equation \(\frac{2}{3}\ x=18\) by \(\frac{2}{3}\) to isolate \(x.\) While this would work, multiplying by the reciprocal requires fewer steps.
Solve: \(\frac{2}{5}\ n=14.\)

Giải đáp

n = 35
Solve: \(\frac{5}{6}\ y=15.\)

Giải đáp

y = 18
Solve: \(8x+9x-5x=-3+15.\)

Giải đáp

Start by combining like terms to simplify each side.
Combine like terms.
Divide both sides by 12 to isolate x.
Simplify.
Check your answer. Let \(x=1\)
Solve: \(7x+6x-4x=-8+26.\)

Giải đáp

x = 2
Solve: \(11n-3n-6n=7-17.\)

Giải đáp

n = −5
Solve: \(11-20=17y-8y-6y.\)

Giải đáp

Simplify each side by combining like terms.
Simplify each side.
Divide both sides by 3 to isolate y.
Simplify.
Check your answer. Let \(y=-3\)
Notice that the variable ended up on the right side of the equal sign when we solved the equation. You may prefer to take one more step to write the solution with the variable on the left side of the equal sign.
Solve: \(18-27=15c-9c-3c.\)

Giải đáp

c = −3
Solve: \(18-22=12x-x-4x.\)

Giải đáp

\(x=-\frac{4}{7}\)
Solve: \(-3(n-2)-6=21.\)

Giải đáp

Remember—always simplify each side first.
Distribute.
Simplify.
Divide both sides by -3 to isolate n.

Check your answer. Let \(n=-7\).
Solve: \(-4(n-2)-8=24.\)

Giải đáp

n = −6
Solve: \(-6(n-2)-12=30.\)

Giải đáp

n = −5
\(-16p=-64\)
\(0.25z=3.25\)
\(0.75a=11.25\)

Giải đáp

a = 15
\(\frac{x}{4}=15\)
\(\frac{z}{2}=14\)

Giải đáp

z = 28
\(-20=\frac{\ q}{-5}\)
\(\frac{\ c}{-3}=-12\)

Giải đáp

c = 36
\(\frac{y}{9}=-6\)
\(\frac{q}{6}=-8\)

Giải đáp

q = −48
\(\frac{m}{-12}=5\)
\(-4=\frac{p}{-20}\)

Giải đáp

p = 80
\(\frac{2}{3}\ y=18\)
\(\frac{3}{5}\ r=15\)

Giải đáp

r = 25
\(-\frac{5}{8}\ w=40\)
\(24=-\frac{3}{4}\ x\)

Giải đáp

x = −32
\(-\frac{2}{5}=\frac{1}{10}\ a\)

Symbols used here

\neq

not equal
The two sides are different.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Solve Equations Using the Division and Multiplication Properties of Equality

Solve equations using the Division and Multiplication Properties of Equality
Solve equations that need to be simplified

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Thử đi.

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

More in Arithmetic

Order of operations Fractions Percentages Powers and roots


Multiply by the reciprocal of \(\frac{2}{3}\).
Reciprocals multiply to one.
Multiply.
Check your answer. Let \(x=27\)