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Solve Linear Equations with Variable on both Sides

  • Last Updated : 27 Oct, 2020
Geek Week

Equations contain two types of quantities, one variable the other is number. But we can consider both equivalents for the sake of reverse operations. This is one of the important things that a student must keep in mind for solving various problems. Also, a good understanding of reverse operations would lead to a better level of problem solving.

In some questions, the variable will appear on both sides of an equation. To solve this kind of questions we need to note the following points: 

  • We can add a number with a variable to both sides without changing the equation or the values.
  • We can subtract a number with a variable to both sides without changing the equation or the values.
  • We can multiply a number with a variable to both sides without changing the equation or the values.
  • We can divide a number with a variable to both sides without changing the equation or the values.

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

Step 1: First, we need separate variables on one side and number on 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 co-efficient 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

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 + 18y = 19 – 18y + 18y

51y – 32 = 19 

51y – 32 + 32 = 19 + 32

51y = 51

y = 1 

Check: 33y – 32 = 19 – 18y? 

Yes!

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