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Unit 7: Lesson 4
- One-step division equations
- One-step multiplication equations
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
Let's try solving 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
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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.
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Click the equation to see how to solve it.
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A complete, free online christian homeschool curriculum for your family and mine, solving equations by multiplying or dividing.
Multiplication and division are inverse operations. If an equation contains multiplication, solve it by dividing both sides to undo the multiplication. Similarly, if an equation contains division, solve it by multiplying both sides to undo the division.
Solving Multiplication Equations
To solve a multiplication equation, one must divide the same number from each side of the equation so that the variable will be by itself on one side of the equation and the answer on the other side.
Solving Division Equations
To solve a division equation, one must multiply the same number on each side of the equation so that the variable will be by itself on one side of the equation and the answer on the other side.
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Chapter Contents ⊗
- 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
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Later, on this page
Dividing by a fraction.
- Long Division
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)`
Next, cancel the numbers top and bottom (we divide top and bottom by `4`), the " a " terms (we cancel `a^2=aa` from top and bottom) and the " b " terms (we cancel `b^3=b b b` from top and bottom) to give us the final answer:
`(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)`
With this example, we'll break it into 2 fractions, both with denominator 4 q to make it easier to see what to do.
`(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`
After you have had some practice with these, you'll be able to do it without separating them into 2 fractions first.
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)`
I'll show you how to do this two different ways. It is worth seeing both, because they are both useful. You can decide which is easier ;-)
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`
So the question has become:
`(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
I recognise that I have "/ x " in both the numerator and denominator. So if I just multiply top and bottom by x , it will simplify everything by removing the fractions on top and bottom.
`(3+1/x)/(5/x+4)xxx/x`
I am really just multiplying by "1" and not changing the original value of the fraction - just changing its form.
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.
Now we have
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)`
We are dividing a polynomial of degree 2 by a polynomial of degree 1. This is algebraic long division.
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
We write 3 x at top of our long division and multiply (3 x )( x − 4) = 3 x 2 − 12 x to give the second row of our solution.
Step 2: Subtracting the second row from the first gives:
Be careful with
-11 x − (-12 x ) = -11 x + 12 x = x
Step 3: Bring down the -4 from the first row:
Step 4: Divide x (in the 3rd row) by x from the ( x − 4) in the question. Our answer is 1 and we write "+ 1" at the top of our long division.
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) .
Simplify `(6x^2+6+7x)/(2x+1)`
We can think of `(6x^2+6+7x)/(2x+1)` as (6 x 2 + 7 x + 6) ÷ (2 x + 1)
Once again we are dividing a polynomial of degree 2 by a polynomial of lower degree (1). This is algebraic long division.
Step 1: 6 x 2 ÷ 2 x = 3 x
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 3: Bring down the 6 :
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.
So the answer is:
`(6x^2+6+7x)/(2x+1)=3x+2+4/(2x+1)`
NOTE: Some people prefer to write the problem with all the x 2 's, x 's and units in line, as follows:
You can see how algebraic long division is used in a later section, Remainder and Factor Theorems .
Helping with Math
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Algebra Worksheet: Division Equations
Related resources.
The various resources listed below are aligned to the same standard, (7EE03) taken from the CCSM (Common Core Standards For Mathematics) as the Expressions and equations Worksheet shown above.
Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.
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
Similar to the above listing, the resources below are aligned to related standards in the Common Core For Mathematics that together support the following learning outcome:
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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Send us your math problem and we'll help you solve it - right now.
Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals)