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

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 #2

Solve for X:

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

Video Solution

Step-by-Step Solution

To solve the equation 13x=9\frac{1}{3}x = 9, we need to isolate the variable xx. To accomplish this, we can multiply both sides of the equation by 3, the reciprocal of 13\frac{1}{3}.

Step-by-step solution:

  • Step 1: Multiply both sides by 3.
    (3×13)x=3×9\left(3 \times \frac{1}{3}\right)x = 3 \times 9
  • Step 2: Simplify the left side.
    This gives us 1x=271x = 27, since (3×13)=1\left(3 \times \frac{1}{3}\right) = 1.
  • Step 3: Conclude that x=27x = 27.

Therefore, the solution to the equation is x=27 x = 27 . This matches choice number 1 from the provided options.

Answer

27

Exercise #3

Solve for X:

6x=72 6x=72

Video Solution

Step-by-Step Solution

To solve for xx in the equation 6x=726x = 72, follow these steps:

Step 1: Identify the equation and the coefficient of xx.
The given equation is 6x=726x = 72, where the coefficient of xx is 6.

Step 2: Isolate xx by dividing both sides of the equation by the coefficient (6).
Perform the division: x=726x = \frac{72}{6}.

Step 3: Simplify the result.
Calculating 726\frac{72}{6}, we get x=12x = 12.

Therefore, the solution to the equation is x=12x = 12.

Answer

12

Exercise #4

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 #5

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=18×14=1×18×4=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 #6

6x=18 -6x=18

Video Solution

Step-by-Step Solution

To solve the equation 6x=18-6x = 18, we need to isolate the variable xx.

Our equation is:

6x=18-6x = 18

The variable xx is multiplied by 6-6. To undo this operation and solve for xx, we divide both sides of the equation by 6-6. This will isolate xx on one side of the equation:

6x6=186\frac{-6x}{-6} = \frac{18}{-6}

Simplifying both sides, we find:

x=3x = -3

Thus, the solution to the equation 6x=18-6x = 18 is x=3x = -3.

Therefore, the correct answer is x=3x = -3.

Answer

3 -3

Exercise #7

7y=27 -7y=-27

Video Solution

Step-by-Step Solution

To solve the equation 7y=27-7y = -27, we need to isolate the variable yy. We do this by performing the following steps:

  • Step 1: Divide both sides of the equation by 7-7 to solve for yy.

Performing this operation gives us:

7y÷(7)=27÷(7)-7y \div (-7) = -27 \div (-7)

Simplifying both sides, we have:

y=277y = \frac{27}{7}

To express 277\frac{27}{7} as a mixed number, we divide 27 by 7:

  • 27 divided by 7 equals 3 with a remainder of 6. Hence, 277=367\frac{27}{7} = 3\frac{6}{7}.

Therefore, the solution to the equation is y=367y = 3\frac{6}{7}.

Among the given choices, option 1 matches our result.

Therefore, the solution to the problem is y=367 y = 3\frac{6}{7} .

Answer

367 3\frac{6}{7}

Exercise #8

Solve for X:

15x=12 \frac{1}{5}x=12

Video Solution

Step-by-Step Solution

To solve this problem, we will follow the steps outlined below:

  • Step 1: Recognize that 15x=12 \frac{1}{5}x = 12 gives us x x multiplied by 15 \frac{1}{5} .
  • Step 2: Multiply both sides of the equation by 5 to eliminate the fraction.
  • Step 3: Simplify the resulting equation to solve for x x .

Let's proceed step-by-step:

Step 1: We have the equation 15x=12 \frac{1}{5}x = 12 .

Step 2: To isolate x x , multiply both sides of the equation by 5:

5×15x=5×12 5 \times \frac{1}{5}x = 5 \times 12

Step 3: Simplify both sides:

  • The left side simplifies to x x because 5×15=1 5 \times \frac{1}{5} = 1 , so x x is left alone.
  • The right side becomes 60 60 , since 5×12=60 5 \times 12 = 60 .

Therefore, the value of x x is 60 60 .

Therefore, the solution to the problem is x=60 x = 60 .

Answer

60 60

Exercise #9

Solve for X:

7x=4 7x=4

Video Solution

Step-by-Step Solution

To solve the equation 7x=4 7x = 4 , we will follow these steps:

  • Step 1: We start with the equation 7x=4 7x = 4 .

  • Step 2: Our goal is to isolate x x . Since x x is multiplied by 7, we will divide both sides of the equation by 7.

  • Step 3: Performing division: x=47 x = \frac{4}{7}

Therefore, the solution to the equation 7x=4 7x = 4 is x=47 x = \frac{4}{7} .

Answer

47 \frac{4}{7}

Exercise #10

Solve for X:

6x=3 6x=3

Video Solution

Step-by-Step Solution

To solve the equation 6x=3 6x = 3 , follow these steps:

Step 1: We aim to isolate x x . Divide both sides of the equation by 6 to remove the coefficient attached to x x :

x=36 x = \frac{3}{6}

Step 2: Simplify the fraction on the right side:

x=12 x = \frac{1}{2}

Therefore, the solution to the equation 6x=3 6x = 3 is x=12 x = \frac{1}{2} .

Answer

12 \frac{1}{2}

Exercise #11

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:

3×13x=3×19 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 #12

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 #13

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 #14

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 #15

Solve for X:

8x=5 8x=5

Video Solution

Step-by-Step Solution

To solve the equation 8x=5 8x = 5 , follow these steps:

  • Step 1: Identify the equation 8x=5 8x = 5 , where x x is the unknown variable.
  • Step 2: To isolate x x , divide both sides of the equation by 8.
    This step involves equivalent operations to maintain equality.
  • Step 3: Perform the division on both sides:
    8x8=58\frac{8x}{8} = \frac{5}{8}.
    This simplifies to x=58 x = \frac{5}{8} .

Now, let's outline these steps in detail:

We begin with the equation 8x=5 8x = 5 .

Dividing both sides by the coefficient of x x , which is 8, gives:

8x8=58\frac{8x}{8} = \frac{5}{8}.

This simplifies directly to:

x=58 x = \frac{5}{8} .

Therefore, the solution to the problem is x=58 x = \frac{5}{8} .

Answer

58 \frac{5}{8}