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## Unit 7: Lesson 4

## One-step multiplication & division equations

- One-step multiplication & division equations: fractions & decimals
- One-step multiplication equations: fractional coefficients

## Multiplication and division are inverse operations

If we start with 7, multiply by 3, then divide by 3, we get back to 7:

7 ⋅ 3 ÷ 3 = 7 7 \cdot 3 \div 3 = 7 7 ⋅ 3 ÷ 3 = 7 7, dot, 3, divided by, 3, equals, 7

If we start with 8, divide by 4, then multiply by 4, we get back to 8:

8 ÷ 4 ⋅ 4 = 8 8 \div 4 \cdot 4 = 8 8 ÷ 4 ⋅ 4 = 8 8, divided by, 4, dot, 4, equals, 8

## Solving a multiplication equation using inverse operations

6 t = 54 6 t 6 = 54 6 Divide each side by six. t = 9 Simplify. \begin{aligned} 6t &= 54 \\\\ \dfrac{6t}{\blueD{6}} &= \dfrac{54}{\blueD{ 6}}~~~~~~~~~~\small\gray{\text{Divide each side by six.}} \\\\ t &= \greenD{9}~~~~~~~~~~\small\gray{\text{Simplify.}} \end{aligned} 6 t 6 6 t t = 5 4 = 6 5 4 Divide each side by six. = 9 Simplify.

## Let's check our work.

Solving a division equation using inverse operations.

x 5 = 7 x 5 ⋅ 5 = 7 ⋅ 5 Multiply each side by five. x = 35 Simplify. \begin{aligned} \dfrac x5 &= 7 \\\\ \dfrac x5 \cdot \blueD{5} &= 7 \cdot \blueD{5}~~~~~~~~~~\small\gray{\text{Multiply each side by five.}} \\\\ x &= \greenD{35}~~~~~~~~~~\small\gray{\text{Simplify.}} \end{aligned} 5 x 5 x ⋅ 5 x = 7 = 7 ⋅ 5 Multiply each side by five. = 3 5 Simplify.

## Summary of how to solve multiplication and division equations

- (Choice A) Multiply each side by 8 8 8 8 . A Multiply each side by 8 8 8 8 .
- (Choice B) Divide each side by 8 8 8 8 . B Divide each side by 8 8 8 8 .
- (Choice C) Multiply each side by 72 72 7 2 72 . C Multiply each side by 72 72 7 2 72 .
- (Choice D) Divide each side by 72 72 7 2 72 . D Divide each side by 72 72 7 2 72 .
- Your answer should be
- an integer, like 6 6 6 6
- a simplified proper fraction, like 3 / 5 3/5 3 / 5 3, slash, 5
- a simplified improper fraction, like 7 / 4 7/4 7 / 4 7, slash, 4
- a mixed number, like 1 3 / 4 1\ 3/4 1 3 / 4 1, space, 3, slash, 4
- an exact decimal, like 0.75 0.75 0 . 7 5 0, point, 75
- a multiple of pi, like 12 pi 12\ \text{pi} 1 2 pi 12, space, start text, p, i, end text or 2 / 3 pi 2/3\ \text{pi} 2 / 3 pi 2, slash, 3, space, start text, p, i, end text

## Want to join the conversation?

## Solving Algebra Equations with Multiplication and Division

- Always perform the same operation to both sides of the equation.
- When you multiply or divide, you have to multiply and divide by the entire side of the equation.
- Try to perform addition and subtraction first to get some multiple of x by itself on one side.
- Always double check you answer by plugging it back into the original equation.

Click the equation to see how to solve it.

## Easy Peasy All-in-One Homeschool

## Solving Multiplication Equations

## Solving Division Equations

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## Tips, tricks, lessons, and tutoring to help reduce test anxiety and move to the top of the class.

## Want Better Math Grades?

- Basic Algebra
- 1. Addition and Subtraction of Algebraic Expressions
- 2. Multiplication of Algebraic Expressions

## 3. Division of Algebraic Expressions

- 4. Solving Equations
- 5. Formulas and Literal Equations
- 6. Applied Verbal Problems
- al-Khwarizmi Father of Algebra

Need help? Chat with a tutor anytime, 24/7 .

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## Later, on this page

Our first examples of division of algebraic expressions involve simplifying and canceling.

Simplify `(3ab(4a^2b^5))/(8a^2b^3)`

First, we multiply out the top line:

`(12a^3b^6)/(8a^2b^3)`

When we write it out in full, this means

`(12 xx aaa xx b b b b b b)/(8 xx aa xx b b b)`

`(3ab^3)/2`

Simplify `(12m^2n^3)/((6m^4n^5)^2)`

We square the denominator (bottom) of the fraction:

`(12m^2n^3)/((6m^4n^5)^2)=(12m^2n^3)/(36m^8n^10)`

Next, we cancel out the numbers, and the " m " and " n " terms to give the final answer:

`1/(3m^6n^7)`

Simplify `(6p^3q^2-10p^2q)/(4q)`

`(6p^3q^2-10p^2q)/(4q)=(6p^3q^2)/(4q)-(10p^2q)/(4q)`

Next, we cancel the numbers and variables:

`(3p^3q)/2-(5p^2)/2`

Finally, we combine the fractions:

`(3p^3q-5p^2)/2`

Recall the following when dividing algebraic expressions.

The reciprocal of a number x , is `1/x`.

For example, the reciprocal of 5 is `1/5` and the reciprocal of `1 2/3` is `3/5`.

To divide by a fraction, you multiply by the reciprocal of the fraction.

For example, `3/4 -: 7/x=3/4xxx/7=(3x)/28`

`(3+1/x)/(5/x+4)`

## Solution 1 - Multiplying by the Reciprocal

I take the top expression (numerator) and turn it into a single fraction with denominator x .

`3+1/x=(3x+1)/x`

We do likewise with the bottom expression (denominator):

`5/x+4=(5+4x)/x`

`(3+1/x)/(5/x+4)=((3x+1)/x)/((5+4x)/x)`

We think of the right side as a division of the top by the bottom:

`(3x+1)/x-:(5+4x)/x`

To divide by a fraction, you multiply by the reciprocal:

`(3x+1)/(x)xxx/(5+4x)=(3x+1)/(5+4x)`

The x 's cancelled out, and we have our final answer, which cannot be simplified any more.

## Solution 2 - Multiplying Top and Bottom

`(3+1/x)/(5/x+4)xxx/x`

So I multiply each element of the top by x and each element of the bottom by x and I get:

`(3+1/x)/(5/x+4)xxx/x=(3x+1)/(5+4x)`

I cannot simplify any further.

## Long Division in Algebra

Before we do an example using algebra, let’s remember how to do long division with numbers first.

Let’s do 23,576 divided by 13.

We can write this as a fraction:

Now, to divide this, (assuming we do not have a calculator) we could proceed as follows.

23 divided by 13 = 1 with remainder 10.

We bring the 5 (the next number after 3) down.

105 divided by 13 is 8 with remainder 1

We continue until we get to the last number, 6.

Our result means that the answer is 1,813 with remainder 7, or:

`23576/13=1813 7/13`

We use a similar technique for long division in algebra.

## Example 6 - Algebraic Long Division

Simplify `(3x^2-11x-4)-:(x-4)`

Step 1: We look at the first term of (3 x 2 − 11 x − 4) and the first term of ( x − 4) .

Divide as follows: 3 x 2 ÷ x = 3 x

Step 2: Subtracting the second row from the first gives:

-11 x − (-12 x ) = -11 x + 12 x = x

Step 3: Bring down the -4 from the first row:

Next, multiply (1) by ( x − 4) to get the 4th row.

Step 5: Subtract the 4th row from the 3rd:

So (3 x 2 − 11 x − 4) ÷ ( x − 4) = 3 x + 1

You can check your answer by multiplying (3 x + 1) by ( x − 4) and you'll get (3 x 2 − 11 x − 4) .

We can think of `(6x^2+6+7x)/(2x+1)` as (6 x 2 + 7 x + 6) ÷ (2 x + 1)

So we write the following, using (3 x )(2 x + 1) = 6 x 2 + 3 x for the second row:

Step 2: We subtract 6 x 2 + 3 x from the first row:

Step 4: Divide 4 x by 2 x . Our answer is 2 and we multiply 2(2 x + 1) to get the 4th row.

Step 5: Subtract, and we are left with 4.

`(6x^2+6+7x)/(2x+1)=3x+2+4/(2x+1)`

You can see how algebraic long division is used in a later section, Remainder and Factor Theorems .

## Algebra Worksheet: Division Equations

## Simple Equations

- Equations with subtraction e.g. x – 4 = 2
- Subtraction & addition equations (1 of 2) e.g. a + 3 = 7 and x – 9 = 11
- Subtraction & addition equations (2 of 2) e.g. a + 3 = 7 and x – 9 = 11
- Multiplication & division equations (1 of 2) e.g. 3n = 12 and a/7 = 3
- Multiplication & division equations (2 of 2) e.g. 3n = 12 and a/7 = 3
- Addition, subtraction, multiplication & division equations
- Solving equations in two steps (1 of 4) e.g. 5n + 4 = 29
- Solving equations in two steps (2 of 4) e.g. a/4 + 3 = 7
- Solving equations in two steps (3 of 4) e.g. 7n – 3 = 18
- Solving equations in two steps (4 of 4) e.g. b/9 – 4 = 6

Solve real-life and mathematical problems using numerical and algebraic expressions and equations

- Solving Simple Equations in 2 Steps (From Example/Guidance)
- Inequalities for word problems (2 pages) (From Worksheet)

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