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How to Solve Equations With Variables on Both Sides

Solving Equations With Variables Both on Sides 

Have you ever been stuck while solving equations with variables in your exam or math class? It is a common challenge that students usually face. 

Solving equations with variables on both sides requires a systematic approach to consolidate all terms containing the variable on one side and all constant numbers on the other. 

In this blog, you will learn how to do equations with variables on both sides.

Equations With Variables on Both Sides

Equations with variables on both sides are linear equations where the unknown value (often x) appears on both the left-hand side (LHS) and right-hand side (RHS) of the equals sign. 

The primary goal in solving these is to isolate the variable by moving all variable terms to one side and all constant terms to the other. 

Examples 

6x+7=3x-8

2x+9=3(x-4)

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Solving Equations With Variables on Both Sides

To solve an equation with variables on both sides, the goal is to isolate the variable on one side and the constant numbers on the other. You will learn to solve equations with variables on both sides step by step. 

Steps to solve variables on both sides

  • Simplify each side: Use the distributive property to remove parentheses and combine like terms on each side of the equals sign individually.
  • Move the variables to one side: Use addition or subtraction (inverse operations) to move all terms containing the variable to one side. It is often easiest to move the variable with the smaller coefficient to avoid working with negative numbers.
  • Move the constants to the other side: Once the variables are on one side, move the constants to the opposite side using inverse operations.
  • Isolate the variable: Divide or multiply both sides by the coefficient of the variable to find the final value.
  • Check your answer: Substitute your result back into the original equation to ensure both sides balance

Solving Simple Equations with Variables on Both Sides 

To solve the simple equations that do not contain parentheses or fractions, we isolate the variable using the inverse operation. Here is the step-by-step guidance with an example. 

Example 

Solve 3x-8=2x-6

This is a simple equation with no fractions and parentheses. To solve this, we have to get all the constants on the right-hand side of the equation. 
To isolate the variable on the left side of the equation, we add 8 to both sides. 

3x-8+8=2x-6+8

3x=2x+2

To eliminate 2x from right hand side, we will subtract it from both sides.

3x-2x=2x-2x+2

Now, we will simplify the equation.

x=2

Verify

To verify the answer, substitute the value of x in the original equation.

3(2)-8=2(2)-6

6-8=4-6

-2=-2

Both sides are equal. So, our answer x=2 is correct. 

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Solving Equations with Variables on Both Sides with Parentheses

To solve equations with variables on both sides with parentheses, we take the help of the distributive property. Here is the step by step guide.

  • Use the distributive property to multiply coefficients outside the parentheses by each term inside.
  • Simplify each side of the equation by adding or subtracting like terms.
  • Use inverse operations (addition or subtraction) to move all variable terms to one side of the equation.
  • Move numerical constants to the opposite side of the variables.
  • Use multiplication or division to solve for the variable

Example 

Solve the equation 2(3x-4)+5=5(2x+3)-2

Use the distributive property to multiply coefficients outside the parentheses.

 2(3x-4)+5=5(2x+3)-2

6x-8+5=10x+15-2

Combine the like terms

6x-3=10x+13

To get constant on one side, add 3 to both sides.

6x-3+3=10x+13+3
6x=10x+16

Now subtract 10x from both sides.

6x-10x=10x-10+16

-4x=16

x=16-4

x=-4
Verify 

To verify the answer, substitute the value of x in the original equation.

2(3(-4) - 4) + 5=5(2(-4) + 3) - 2

2(-12 - 4) + 5=5(-8 + 3) - 2

2(-16) + 5=5(-5) - 2

-32 + 5=-25 - 2

-27=-27

Since both sides are equal. Our answer x= -4 is correct. 

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Solving Equations with Variables on Both Sides with Fractions

To solve equations with variables on both sides with fractions, we clear the fractions by multiplying all terms by a common denominator. Then solve the equation.

Here is the step by step guidance.

  • Find the Least Common Denominator (LCD) of all fractions appearing on both sides of the equation.
  • Multiply both the left and right sides of the equation by this number. This eliminates all denominators, leaving only integers.
  • If there are parentheses, use the distributive property to expand them and combine any like terms on each side.
  • Use addition or subtraction to move all terms containing the variable to one side.
  • Use addition or subtraction to move all constant numbers to the opposite side.
  • Divide both sides by the coefficient of the variable to find the final value.

Example 

Example

Solve 3/5x - 5 = 6 - 3/6x

First, we will find the least common denominator (LCD) of the fractions. The LCD of 5 and 6 is 30.

Multiply both sides of the equation by 30:

30⋅(35x−5)=30⋅(6−36x)30 \cdot \left(\frac{3}{5}x - 5\right) = 30 \cdot \left(6 - \frac{3}{6}x\right)
18x−150=180−15x18x - 150 = 180 - 15x

Add 150 on both sides:

18x−150+150=180+150−15x18x - 150 + 150 = 180 + 150 - 15x
18x=330−15x18x = 330 - 15x

Add 15x on both sides:

18x+15x=330−15x+15x18x + 15x = 330 - 15x + 15x
33x=33033x = 330
x=33033x = \frac{330}{33}
x=10x = 10

Verify:

For verification, substitute the value of x in the original equation:

35(10)−5=6−36(10)\frac{3}{5}(10) - 5 = 6 - \frac{3}{6}(10)
6−5=6−56 - 5 = 6 - 5
1=11 = 1

Since both sides are equal, the answer x = 10 is correct.

Solving Equations With Variables on Both Sides Worksheets

A worksheet on Math equations with variables is an educational tool to help students practice math problems with variables on both sides. It assists them in understanding the concept and solving the problems easily and confidently. 

We will provide you with solving equations with variables on both sides worksheet PDF. You can download the worksheet and practice at home. 

Download PDF

Conclusion 

Solving equations with variables on both sides involves isolating the variable on one side by consolidating terms through addition or subtraction, simplifying through distribution, and combining like terms. Then, solving the resulting linear equation. Common strategies include moving variables to the left and constants to the right. 

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