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Solving Equations with Variables on Both Sides

An equation is an expression with "equal" sign containing variables and constants joined together with the help of arithmetic operators. The variables in an equation can occur either on one side or on both the sides of "equal" sign.
For example: 3x - 4 = 5x and 7(y - 2) = 1 + 2y etc.
For solving this type of equations, we need to take the variables on one side.

There are certain rules for solving equations with variables on both the sides which are as follows:

• We can add or subtract a number or a variable on both the sides.
• We can multiply or divide all the terms on both the sides by a number or a variable.
• While changing the side of a variable or a constant, its sign is reversed. That means, positive quantity becomes negative and negative quantity become positive.
• When we change the side, multiplication operation becomes division and division operation becomes multiplication.

Let us look at few examples:

Example 1:

Solve 5x - 8 = 3 + 6x

Solution:

5x - 8 = 3 + 6x
5x - 6x = 3 + 8
- x = 11
x = - 11

Example 2:

Solve 3(t - 1) = 2t + 5

Solution:

3(t - 1) = 2t + 5
3t - 3 = 2t + 5
3t - 2t = 5 + 3
t = 8

Example 3:

$\frac{x+1}{2}=\frac{x-2}{3}$

Solution:

$\frac{x+1}{2}=\frac{x-2}{3}$

By cross multiplication, we get
3(x + 1) = 2 (x - 2)
3x + 3 = 2x - 4
3x - 2x = - 4 - 3
x = - 7

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