Solution of an equation by multiplying/dividing both sides - Examples, Exercises and Solutions

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. 

Solving Equations by Multiplying or Dividing Both Sides by the Same Number

Suggested Topics to Practice in Advance

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

Practice Solution of an equation by multiplying/dividing both sides

Exercise #1

Solve the equation

20:4x=5 20:4x=5

Video Solution

Step-by-Step Solution

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

204x=5 \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

Answer

x=1 x=1

Exercise #2

Solve the equation

5x15=30 5x-15=30

Video Solution

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

Answer

x=9 x=9

Exercise #3

Solve for X:

3x=18 3x=18

Video Solution

Step-by-Step Solution

We use the formula:

ax=b a\cdot x=b

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

Note that the coefficient of X is 3.

Therefore, we will divide both sides by 3:

3x3=183 \frac{3x}{3}=\frac{18}{3}

Then divide accordingly:

x=6 x=6

Answer

6 6

Exercise #4

What is the missing number?

2312×(6)+? ⁣:7=102 23-12\times(-6)+?\colon7=102

Video Solution

Step-by-Step Solution

First, we solve the multiplication exercise:

12×(6)=72 12\times(-6)=-72

Now we get:

23(72)+x ⁣:7=102 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 ⁣:7=102 23+72+x\colon7=102

We reduce:

95+x:7=102 95+x:7=102

We move the sections:

x:7=10295 x:7=102-95

x:7=7 x:7=7

x7=7 \frac{x}{7}=7

Multiply by 7:

x=7×7=49 x=7\times7=49

Answer

49

Exercise #5

Find the value of the parameter X

13x+56=16 \frac{1}{3}x+\frac{5}{6}=-\frac{1}{6}

Video Solution

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 56 \frac{5}{6} to the other side, and we will get

13x=1656 \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:

 13x=66 \frac{1}{3}x=-\frac{6}{6}

Observe the minus sign on the right side!

 

13x=1 \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 1x=-3

 x=3 x=-3

Answer

-3

Exercise #1

Solve for X:

5x=38 5x=\frac{3}{8}

Video Solution

Step-by-Step Solution

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

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

Answer

340 \frac{3}{40}

Exercise #2

Solve for X:

x4=3 \frac{x}{4}=3

Video Solution

Step-by-Step Solution

We use the formula:

ax=b a\cdot x=b

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

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

x4=3 \frac{x}{4}=3

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

4×x4=3×4 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 x=12

Answer

12 12

Exercise #3

Solve for X:

x+43=78 \frac{x+4}{3}=\frac{7}{8}

Video Solution

Step-by-Step Solution

First, we cross multiply:

8×(x+4)=3×7 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 8x+32=21

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

8x=2132 8x=21-32
Solve the subtraction exercise on the right side and divide by 8:

8x=11 8x=-11

8x8=118 \frac{8x}{8}=-\frac{11}{8}

Convert the simple fraction into a mixed fraction:

x=138 x=-1\frac{3}{8}

Answer

138 -1\frac{3}{8}

Exercise #4

92x×224 ⁣:4=64 92-x\times2-24\colon4=64 Calculate X.

Video Solution

Step-by-Step Solution

First, we solve the multiplication and division exercises, we will put them in parentheses to avoid confusion:

92(x×2)(24 ⁣:4)=64 92-(x\times2)-(24\colon4)=64

922x6=64 92-2x-6=64

Reduce:

862x=64 86-2x=64

Move the sides:

2x=6486 -2x=64-86

2x=22 -2x=-22

Divide by negative 2:

x=222 x=\frac{-22}{-2}

x=11 x=11

Answer

11

Exercise #5

Solve the equation

413x=2123 4\frac{1}{3}\cdot x=21\frac{2}{3}

Video Solution

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.

Let's start with the four and the third:

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

Answer

x=5 x=5

Exercise #1

Solve for X:

18x=34 \frac{1}{8}x=\frac{3}{4}

Video Solution

Step-by-Step Solution

We use the formula:

abx=cd \frac{a}{b}x=\frac{c}{d}

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

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

x8=34 \frac{x}{8}=\frac{3}{4}

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

8×x8=34×8 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=244=6 x=\frac{24}{4}=6

Answer

6 6

Exercise #2

Solve for x:

8x45=2x+24 \frac{8x-4}{5}=\frac{2x+2}{4}

Video Solution

Step-by-Step Solution

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

4(8x4)=5(2x+2) 4(8x-4)=5(2x+2)

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

32x16=10x+10 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:

32x10x=10+16 32x-10x=10+16

We calculate the elements:

22x=26 22x=26

We divide the two sections by 22:

22x22=2622 \frac{22x}{22}=\frac{26}{22}

x=2622 x=\frac{26}{22}

Answer

2622 \frac{26}{22}

Exercise #3

2b3b+4=5 2b-3b+4=5

b=? b=\text{?}

Video Solution

Answer

-1

Exercise #4

x+2x=9 x+2x=9

x=? x=\text{?}

Video Solution

Answer

3

Exercise #5

6x=18 -6x=18

Video Solution

Answer

3 -3

Topics learned in later sections

  1. Solving Equations by Simplifying Like Terms
  2. Solving Equations Using the Distributive Property
  3. First-degree equations with one unknown