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

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:

Identify the Coefficient: Determine if multiplication or division is needed to isolate the variable.
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.

Detailed explanation

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Test yourself on solving an equation by multiplication/ division!

$$ 4x:30=2 $$

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

Example 1

$3X=24$

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

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

$3X=24$ / $:3$

$X=8$

The result of the equation is $8$ .

Example 2

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

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.

$\frac{X}{2}=5$ / $\times2$

$X=10$

The result of the equation is $10$ .

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

Exercise #1

$4x:30=2$

Video Solution

Step-by-Step Solution

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

Step 1: Recognize that $4x:30$ implies $\dfrac{4x}{30} = 2$ .
Step 2: Eliminate the fraction by multiplying both sides of the equation by 30.
Step 3: Simplify the equation to solve for $x$ .

Now, let's work through each step:

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

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

This simplifies to:
$4x = 60$

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

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

Checking choices, the correct answer is:

$x = 15$

Answer

$x=15$

Exercise #2

Solve the equation

$20:4x=5$

Video Solution

Step-by-Step Solution

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

$\frac{20}{4x}=5$

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

$x=1$

Exercise #3

Solve for X:

$5x=25$

Video Solution

Step-by-Step Solution

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

Divide both sides of the equation by 5:

\frac{5x}{5} = \frac{25}{5}

After performing the division, we get:

x = 5

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

Answer

Exercise #4

Solve for X:

$6x=72$

Video Solution

Step-by-Step Solution

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

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

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

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

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

Answer

Exercise #5

Solve for X:

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

Video Solution

Step-by-Step Solution

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

Step-by-step solution:

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