Solving Linear Equations with Variable on both Sides

Equations consist of two main components: variables and numbers. Understanding the relationship between these components and how to manipulate them is essential for solving equations.

• Variable: A variable is a symbol (often a letter like x, y, or z) that represents an unknown or changing quantity.
• Number: A number is a constant value in an equation.

Methods for Solving Equations with Variables on Both Sides

When a variable appears on both sides of an equation, the following principles apply to simplify and solve the equation:

1. Adding/Subtracting Terms: You can add or subtract the same number (or expression) to both sides of the equation without altering its equality.
2. Multiplying/Dividing Terms: You can multiply or divide both sides of the equation by the same non-zero number without changing its equality.

These operations are called reverse operations, and they help you isolate the variable to solve the equation.

Example: Solve 14 - 2x = 5x for the value of x.

Step 1: First, we need separate variables on one side and numbers on the other side by applying some basic operations.

Add 2x on both the sides
14 - 2x + 2x = 5x + 2x

(Similarly, we can subtract a term with a variable from both sides of the equation)

Step 2: Perform operations to convert the coefficient of the variable to 1.

Equation: 14 = 7x
Divide 7 on both the sides

x = 2

Example: Solve 64 + 2x = 10x + 8 for the value of x

Step 1: Subtract 2x from both sides:

64 + 2x - 2x = 10x - 2x + 8
64 = 8x + 8

Step 2: Subtract 8 from both the sides:

64 - 8 = 8x + 8 - 8
56 = 8x 

Step 3: Divide 8 on both the sides

x = 7

Note: In every problem of this kind it is always recommended separating the numbers and variables on either side of the equation by applying the reverse operations.

Sample Problems on Linear Equations with variables on both Sides

Example 1. Solve for x: 35x - 45 = 25

Solution: 

Add 45 to both the sides 
35x - 45 + 45 = 25 + 45 
35x = 70

Divide 35 on both the sides 
x = 2 

Example 2. Solve for x: 22 - 32x = 33 + x

Solution: 

Add 32x on both the sides 
22 - 32x + 32x = 33 + x + 32x
22 = 33 + 33x

Subtract 33 from both the sides 
22 - 33 = 33 + 33x -33
-11 = 33x

Divide 11 on both the sides
-1 = 3x

Divide 3 on both the sides 
x = -1/3

Example 3. Solve for x: 23x + 4 = 104 + 3x

Solution: 

Subtract 4 from both the sides 23x + 4 - 4 = 104 + 3x - 4
23x = 100 + 3x

Subtract 3x from both the sides
23x - 3x = 100 + 3x - 3x
20x = 100

Divide 20 from both the sides 
x = 5

Example 4. Solve for x: 45x + 21 = 15x + 141

Solution:

Subtract 21 from both the sides 45x + 21 - 21 = 15x + 141 - 21
45x = 15x + 120

Subtract 15x from both the sides 45x - 15x = 15x + 120 - 15x
30x = 120

Divide 30 on both the sides
x = 4

Example 5. Solve for x: 28x + 33 = 108 + 3x

Solution:

Subtract 3x from both the sides 28x + 33 -3x = 108 + 3x - 3x
25x + 33 = 108 

Subtract 33 from both the sides
25x + 33 - 33 = 108 - 33
25x = 75 

Divide 25 on both the sides
x = 3

Example 6. Solve for x: 8x + 3x = 34 + 2 + 2x

Solution:

Simplify: 11x = 36 + 2x

Get the variable on one side: 11x - 2x = 36 + 2x - 2x
9x = 36

Solve using inverse operations:
x = 4

Check Whether: 8(4) + 3(4) = 34 + 2 + 2(4)?  Yes!

Example 7. Solve for y: 33y - 32 = 19 - 18y

Solution:

The equation is already simplified. Get the variable on one side using inverse operations

33y - 32 = 19 - 18y
51y - 32 = 19 
51y = 19 + 32
51y = 51
y = 1

Check: 33y - 32 = 19 - 18y? Yes!

Solve Linear Equations with Variable on both Sides Worksheet

Problem 1: Solve for x: 3x + 5 = 2x + 8

Problem 2: Solve for y: 4y − 2 = 2y + 6.

Problem 3: Solve for x: 5(x − 2) = 2x + 7

Problem 4: Solve for z: 7z + 3 = 5z + 9

Problem 5: Solve for x: 6x − 4 = 2x + 12

Problem 6: Solve for a: 2(a + 3) = 4 + 3a

Problem 7: Solve for b: 8 − 3b = 2b + 1

Problem 8: Solve for m: 3m − 4 = 5(m + 2)

Answer Key

1. x = 3
2. y = 4
3. x = 17/3
4. z = 3
5. x = 4
6. a = 2
7. b = 7/5
8. m = −7