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

๐Ÿ†Practice solving an equation by multiplication/ division

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.ย 

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 coefficientโ€™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.

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Test yourself on solving an equation by multiplication/ division!

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Find the value of the parameter X

\( \frac{1}{3}x=\frac{1}{9} \)

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Below, we provide you with some examples where we apply this method.

Example 1

3X=24 3X=24

We solve the equation and find the numerical value of X X by dividing both sides of the equation by the number 3 3 .

In this way, we neutralize and isolate the X X on the left side of the equation, while on the right side we obtain the result of the equation.

3X=24 3X=24 / :3 :3

X=8 X=8

The result of the equation is 8 8 .


Example 2

X2=5 \frac{X}{2}=5

We solve the equation and find the numerical value of X by multiplying both sides of the equation by the number 2. This way, we neutralize and isolate X on the left side of the equation, while on the right side we obtain the result of the equation.

X2=5 \frac{X}{2}=5 ย / ร—2 \times2

X=10 X=10

The result of the equation is 10 10 .


Examples and exercises with solutions for solving equations by multiplying or dividing both sides by the same number

Exercise #1

Find the value of the parameter X

13x=19 \frac{1}{3}x=\frac{1}{9}

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Identify the given fraction equation.
  • Multiply both sides of the equation by the reciprocal of the coefficient of x x .
  • Simplify to isolate x x .

Now, let's work through these steps:
Step 1: The problem gives us the equation 13x=19 \frac{1}{3} x = \frac{1}{9} .
Step 2: We multiply both sides by 3 to eliminate the fraction on the left side:

3ร—13x=3ร—19 3 \times \frac{1}{3} x = 3 \times \frac{1}{9}

Step 3: Simplifying both sides results in:

x=39 x = \frac{3}{9}

Further simplification of 39\frac{3}{9} yields:

x=13 x = \frac{1}{3}

Therefore, the solution to the problem is 13 \frac{1}{3} .

Answer

13 \frac{1}{3}

Exercise #2

4x:30=2 4x:30=2

Video Solution

Step-by-Step Solution

To solve the given equation 4x:30=2 4x:30 = 2 , we will follow these steps:

  • Step 1: Recognize that 4x:304x:30 implies 4x30=2\dfrac{4x}{30} = 2.

  • Step 2: Eliminate the fraction by multiplying both sides of the equation by 30.

  • Step 3: Simplify the equation to solve for xx.

Now, let's work through each step:

Step 1: The equation is written as 4x30=2\dfrac{4x}{30} = 2.

Step 2: Multiply both sides of the equation by 30 to eliminate the fraction:
30ร—4x30=2ร—30 30 \times \dfrac{4x}{30} = 2 \times 30

This simplifies to:
4x=60 4x = 60

Step 3: Solve for xx by dividing both sides by 4:
x=604=15 x = \dfrac{60}{4} = 15

Therefore, the solution to the problem is x=15 x = 15 .

Checking choices, the correct answer is:

x=15 x = 15

Answer

x=15 x=15

Exercise #3

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 #4

Solve for X:

5x=3 5x=3

Video Solution

Step-by-Step Solution

To solve the equation 5x=3 5x = 3 , we will isolate x x by using division:

  • Step 1: Recognize that x x is multiplied by 5. To isolate x x , we need to undo this multiplication.
  • Step 2: Divide both sides of the equation by 5. This step uses the Division Property of Equality:

5x5=35\frac{5x}{5} = \frac{3}{5}

Step 3: Simplify both sides. The left side simplifies to x x (because 5x5=x \frac{5x}{5} = x ), and the right side is 35 \frac{3}{5} .

Hence, the solution to the equation 5x=3 5x = 3 is x=35 x = \frac{3}{5} .

Answer

35 \frac{3}{5}

Exercise #5

Solve for X:

3x=18 3x=18

Video Solution

Step-by-Step Solution

We use the formula:

aโ‹…x=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

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