With this method, we can multiply or divide both sides of the equation by the same element without thereby altering the overall value of the equation. This means that the final result of the equation will not be affected because we have multiplied or divided both sides by the same element or number.

### Suggested Topics to Practice in Advance

1. Solving Equations by Adding or Subtracting the Same Number from Both Sides

## Examples with solutions for Solving an Equation by Multiplication/ Division

### Exercise #1

$2b-3b+4=5$

$b=\text{?}$

### Step-by-Step Solution

Let's arrange the equation so that on the left side we have the terms with coefficient b and on the right side the numbers without coefficient b

We'll remember that when we move terms across the equals sign, the plus and minus signs will change accordingly:

$2b-3b=5-4$

Let's solve the subtraction exercise on both sides:

$-1b=1$

Let's divide both sides by minus 1:

$b=-1$

-1

### Exercise #2

Solve for X:

$3x=18$

### Step-by-Step Solution

We use the formula:

$a\cdot x=b$

$x=\frac{b}{a}$

Note that the coefficient of X is 3.

Therefore, we will divide both sides by 3:

$\frac{3x}{3}=\frac{18}{3}$

Then divide accordingly:

$x=6$

$6$

### Exercise #3

Solve the equation

$5x-15=30$

### Step-by-Step Solution

We start by moving the sections:

5X-15 = 30
5X = 30+15

5X = 45

Now we divide by 5

X = 9

$x=9$

### Exercise #4

Solve the equation

$20:4x=5$

### Step-by-Step Solution

To solve the exercise, we first rewrite the entire division as a fraction:

$\frac{20}{4x}=5$

Actually, we didn't have to do this step, but it's more convenient for the rest of the process.

To get rid of the fraction, we multiply both sides of the equation by the denominator, 4X.

20=5*4X

20=20X

Now we can reduce both sides of the equation by 20 and we will arrive at the result of:

X=1

$x=1$

### Exercise #5

$\frac{-y}{5}=-25$

### Step-by-Step Solution

Let's multiply the simple fraction by y:

$\frac{-1}{5}\times y=-25$

Now let's reduce both terms by $-\frac{1}{5}$

$y=\frac{-25}{-\frac{1}{5}}$

Let's multiply the fraction by negative 5:

$y=-25\times(-5)=125$

$y=125$

### Exercise #6

Find the value of the parameter X

$\frac{1}{3}x+\frac{5}{6}=-\frac{1}{6}$

### Step-by-Step Solution

First, we will arrange the equation so that we have variables on one side and numbers on the other side.

Therefore, we will move $\frac{5}{6}$ to the other side, and we will get

$\frac{1}{3}x=-\frac{1}{6}-\frac{5}{6}$

Note that the two fractions on the right side share the same denominator, so you can subtract them:

$\frac{1}{3}x=-\frac{6}{6}$

Observe the minus sign on the right side!

$\frac{1}{3}x=-1$

Now, we will try to get rid of the denominator, we will do this by multiplying the entire exercise by the denominator (that is, all terms on both sides of the equation):

$1x=-3$

$x=-3$

-3

### Exercise #7

Solve for X:

$\frac{x}{4}=3$

### Step-by-Step Solution

We use the formula:

$a\cdot x=b$

$x=\frac{b}{a}$

We multiply the numerator by X and write the exercise as follows:

$\frac{x}{4}=3$

We multiply by 4 to get rid of the fraction's denominator:

$4\times\frac{x}{4}=3\times4$

Then, we remove the common factor from the left side and perform the multiplication on right side to obtain:

$x=12$

$12$

### Exercise #8

Solve for X:

$5x=\frac{3}{8}$

### Step-by-Step Solution

$ax=\frac{c}{b}$

$x=\frac{c}{b\cdot a}$

$\frac{3}{40}$

### Exercise #9

Solve for X:

$\frac{1}{8}x=\frac{3}{4}$

### Step-by-Step Solution

We use the formula:

$\frac{a}{b}x=\frac{c}{d}$

$x=\frac{bc}{ad}$

We multiply the numerator by X and write the exercise as follows:

$\frac{x}{8}=\frac{3}{4}$

We multiply both sides by 8 to eliminate the fraction's denominator:

$8\times\frac{x}{8}=\frac{3}{4}\times8$

On the left side, it seems that the 8 is reduced and the right section is multiplied:

$x=\frac{24}{4}=6$

$6$

### Exercise #10

Solve for X:

$\frac{x+4}{3}=\frac{7}{8}$

### Step-by-Step Solution

First, we cross multiply:

$8\times(x+4)=3\times7$

We multiply the right section and expand the parenthesis, multiplying each of the terms by 8:

$8x+32=21$

We rearrange the equation remembering change the plus and minus signs accordingly:

$8x=21-32$
Solve the subtraction exercise on the right side and divide by 8:

$8x=-11$

$\frac{8x}{8}=-\frac{11}{8}$

Convert the simple fraction into a mixed fraction:

$x=-1\frac{3}{8}$

$-1\frac{3}{8}$

### Exercise #11

Solve the equation

$4\frac{1}{3}\cdot x=21\frac{2}{3}$

### Step-by-Step Solution

We have an equation with a variable.

Usually, in these equations, we will be asked to find the value of the missing (X),

This is how we solve it:

To solve the exercise, first we have to change the mixed fractions to an improper fraction,

So that it will then be easier for us to solve them.

To convert a mixed fraction, we start by multiplying the whole number by the denominator

4*3=12

Now we add this to the existing numerator.

12+1=13

And we find that the first fraction is 13/3

Let's continue with the second fraction and do the same in it:
21*3=63

63+2=65

The second fraction is 65/3

We replace the new fractions we found in the equation:

13/3x = 65/3

At this point, we will notice that all the fractions in the exercise share the same denominator, 3.

Therefore, we can multiply the entire equation by 3.

13x=65

Now we want to isolate the unknown, the x.

Therefore, we divide both sides of the equation by the unknown coefficient -
13.

63:13=5

x=5

$x=5$

### Exercise #12

What is the missing number?

$23-12\times(-6)+?\colon7=102$

### Step-by-Step Solution

First, we solve the multiplication exercise:

$12\times(-6)=-72$

Now we get:

$23-(-72)+x\colon7=102$

Let's pay attention to the minus signs, remember that a negative times a negative equals a positive.

We multiply them one by one to be able to open the parentheses:

$23+72+x\colon7=102$

We reduce:

$95+x:7=102$

We move the sections:

$x:7=102-95$

$x:7=7$

$\frac{x}{7}=7$

Multiply by 7:

$x=7\times7=49$

49

### Exercise #13

Solve for x:

$\frac{8x-4}{5}=\frac{2x+2}{4}$

### Step-by-Step Solution

To get rid of the fraction mechanics, we will cross multiply between the sides:

$4(8x-4)=5(2x+2)$

We expand the parentheses by multiplying the outer element by each of the elements inside the parentheses:

$32x-16=10x+10$

We arrange the sides accordingly so that the elements with the X are on the left side and those without the X are on the right side:

$32x-10x=10+16$

We calculate the elements:

$22x=26$

We divide the two sections by 22:

$\frac{22x}{22}=\frac{26}{22}$

$x=\frac{26}{22}$

$\frac{26}{22}$

### Exercise #14

Given: the length of a rectangle is 3 greater than its width.

The area of the rectangle is equal to 27 cm².

Calculate the length of the rectangle

### Step-by-Step Solution

The area of the rectangle is equal to length multiplied by width.

Let's set up the data in the formula:

$27=3x\times x$

$27=3x^2$

$\frac{27}{3}=\frac{3x^2}{3}$

$9=x^2$

$x=\sqrt{9}=3$

$x=3$

### Exercise #15

The area of the rectangle below is equal to 22x.

Calculate x.

### Step-by-Step Solution

The area of the rectangle is equal to the length multiplied by the width.

Let's list the known data:

$22x=\frac{1}{2}x\times(x+8)$

$22x=\frac{1}{2}x^2+\frac{1}{2}x8$

$22x=\frac{1}{2}x^2+4x$

$0=\frac{1}{2}x^2+4x-22x$

$0=\frac{1}{2}x^2-18x$

$0=\frac{1}{2}x(x-36)$

For the equation to be equal, x needs to be equal to 36

$x=36$