Multiplying or Dividing Both Sides of the Equation

Sometimes when solving equations, we may encounter variables with coefficients, which we need to remove to isolate the variable and find its value.
Exactly for those cases, and many more, we have the ability to multiply or divide both sides of the equation by the same number to maintain balance and solve for the variable.

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. 

In order to so we need to follow these two steps:
  1. Identify the Coefficient: Determine if multiplication or division is needed to isolate the variable.
  2. Apply Operation to Both Sides: Multiply or divide by the coefficient’s reciprocal.
Solving Equations by Multiplying or Dividing Both Sides by the Same Number

It's important to remember that when we multiply or divide both sides of an equation, the equation's balance should remain unchanged. This means we can always reverse the operation to return to the original equation. If reversing leads to a different result, it indicates that an error was made in the calculations.

Suggested Topics to Practice in Advance

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

Practice Solving an Equation by Multiplication/ Division

Examples with solutions for Solving an Equation by Multiplication/ Division

Exercise #1

y5=25 \frac{-y}{5}=-25

Video Solution

Step-by-Step Solution

We begin by multiplying the simple fraction by y:

15×y=25 \frac{-1}{5}\times y=-25

We then reduce both terms by 15 -\frac{1}{5}

y=2515 y=\frac{-25}{-\frac{1}{5}}

Finally we multiply the fraction by negative 5:

y=25×(5)=125 y=-25\times(-5)=125

Answer

y=125 y=125

Exercise #2

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 #3

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 #4

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 #5

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 #6

Solve for x x :

5x3=45 5x \cdot 3 = 45

Step-by-Step Solution

To solve the equation5x3=45 5x \cdot 3 = 45 , follow these steps:

1. First, identify the operation needed to solve forx x . In this case, we have a multiplication equation.

2. Therefore, we divide both sides of the equation by 15 (since 5×3=15 5 \times 3 = 15 ) to isolate x x :

x=4515 x = \frac{45}{15}

3. Calculate x x :

x=3 x = 3

Answer

x=3 x=3

Exercise #7

Solve the equation:

6x2=24 6x \cdot 2 = 24

Step-by-Step Solution

To solve the equation 6x2=24 6x \cdot 2 = 24 , follow these steps:

1. First, identify the operation involved. In this case, it is multiplication.

2. Divide both sides of the equation by 12 (since 6×2=12 6 \times 2 = 12 ) to isolate x x :

x=2412 x = \frac{24}{12}

3. Calculate x x :

x=2 x = 2

Answer

x=2 x=2

Exercise #8

Solve the equation:

5x6=90 5x \cdot 6 = 90

Step-by-Step Solution

To solve the equation 5x6=90 5x \cdot 6 = 90 , start by simplifying the left side of the equation:

Divide both sides by 6 to isolate 5x 5x :

5x=906 5x = \frac{90}{6}

This simplifies to:

5x=15 5x = 15

Next, divide both sides by 5 to solve for x x :

x=155 x = \frac{15}{5}

This gives us:

x=3 x = 3

Answer

x=3 x=3

Exercise #9

Solve the equation:

7x4=56 7x \cdot 4 = 56

Step-by-Step Solution

To solve the equation 7x4=56 7x \cdot 4 = 56 , start by simplifying the right side of the equation:

Divide both sides by 4 to isolate 7x 7x :

7x=564 7x = \frac{56}{4}

This simplifies to:

7x=14 7x = 14

Next, divide both sides by 7 to solve for x x :

x=147 x = \frac{14}{7}

This gives us:

x=2 x = 2

Answer

x=2 x=2

Exercise #10

3xx=8 3x - x = 8

x=? x = \text{?}

Step-by-Step Solution

Start by simplifying the left-hand side of the equation:

3xx=2x 3x - x = 2x

So the equation becomes:

2x=8 2x = 8

To find the value of x x , divide both sides by 2:

x=82 x = \frac{8}{2}

Then simplify the fraction:

x=4 x = 4

Thus, the solution to the equation isx=4 x = 4 .

Answer

4

Exercise #11

Solve for X:

25+75=10x 25 + 75 = 10x

Step-by-Step Solution

To solve for x x , we start with the equation:
25+75=10x 25 + 75 = 10x

The left side simplifies to:
100=10x 100 = 10x

To isolate x x , divide both sides by 10:
10010=x \frac{100}{10} = x

x=10 x = 10 , which simplifies to:
x=5 x = 5

Answer

5

Exercise #12

Solve for X:

10+140=30x 10 + 140 = 30x

Step-by-Step Solution

To solve for x x , we start with the equation:
10+140=30x 10 + 140 = 30x

The left side simplifies to:
150=30x 150 = 30x

To isolate x x , divide both sides by 30:
15030=x \frac{150}{30} = x

x=5 x = 5 , which simplifies to:
x=4 x = 4

Answer

4

Exercise #13

Solve for X:

50+10=2x 50 + 10 = 2x

Step-by-Step Solution

To solve for x x , we start with the equation:
50+10=2x 50 + 10 = 2x

The left side simplifies to:
60=2x 60 = 2x

To isolate x x , divide both sides by 2:
602=x \frac{60}{2} = x

x=30 x = 30

Answer

30

Exercise #14

4x+2x=18 4x + 2x = 18

Solve the equation above for x x .

Step-by-Step Solution

Combine like terms on the left-hand side:

4x+2x=6x 4x + 2x = 6x

The equation becomes:

6x=18 6x = 18

Divide both sides by 6 to solve for x x :

x=186 x = \frac{18}{6}

Simplify the division:

x=3 x = 3

Thus, x=3 x = 3 is the solution to the equation.

Answer

x=3 x = 3

Exercise #15

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

b=? b=\text{?}

Video Solution

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:

2b3b=54 2b-3b=5-4

Let's solve the subtraction exercise on both sides:

1b=1 -1b=1

Let's divide both sides by minus 1:

b=1 b=-1

Answer

-1