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

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

Exercise #3

Solve for X:

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

Video Solution

Step-by-Step Solution

To solve the equation 4x=18 4x = \frac{1}{8} , we need to isolate x x . We do this by dividing both sides of the equation by the coefficient of x x , which is 4:

  • Step 1: Write the original equation: 4x=18 4x = \frac{1}{8} .
  • Step 2: Divide both sides by 4 to solve for x x :

x=184 x = \frac{\frac{1}{8}}{4}

  • Step 3: Simplify the right-hand side by multiplying fractions, recalling that dividing by a number is equivalent to multiplying by its reciprocal:

x=18ร—14=1ร—18ร—4=132 x = \frac{1}{8} \times \frac{1}{4} = \frac{1 \times 1}{8 \times 4} = \frac{1}{32}

Thus, the solution to the equation is x=132 x = \frac{1}{32} .

Answer

132 \frac{1}{32}

Exercise #4

Solve for X:

5x=25 5x=25

Video Solution

Step-by-Step Solution

To solve the equation 5x=255x = 25, we will isolate xx using division:

  • Divide both sides of the equation by 5:
5x5=255 \frac{5x}{5} = \frac{25}{5}

After performing the division, we get:

x=5 x = 5

Thus, the solution to the equation is x=5 x = 5 .

Answer

5

Exercise #5

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}

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