# Solving Linear Equations in One Variable

A linear equation in one variable contains terms of one variable without exponents. A linear equation in one variable is formed by simple operations of addition, subtraction, multiplication and division
Examples:

$x + 4 = 10$
$2a - 3 = 7$
$\frac{y}{3}+5 = -3$

Hence solving a linear equation in one variable involves 'undoing' operations, the operations that reverses the effect of the operations already done.

## How to solve a linear equation in one variable?

Vocabulary :  Equivalent equations
Two equations are said to be equivalent if they have the same solutions.

The process of solving equations will involve in reducing the equation to an equivalent equation with the variable isolated on one side of the equation. To achieve this the following properties of equality are used:
1. Addition Property: Equal quantities can be added on either side of an equation to get an equivalent equation.
2. Subtraction Property:Equal quantities can be subtracted from either side of an equation to get an equivalent equation.
3. Multiplication Property: Either side of an equation can be multiplied by equal quantities to get an equivalent equation.
4. Division Property:Either side of an equation can be divided by equal quantities to get an equivalent equation.

## Solving linear equations - Examples:

Example: One step equation.

Solve the equation for x:   $x + 7 = 15.$
In the given equation the expression on the left side is got by adding 7 to the variable x. Hence to undo this addition and isolate x, 7 has to be subtracted on either side of the equation.
$x + 7 = 15$
$-7 -7$
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$x = 8$
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Example : Two step equation

Solve the equation: $3a - 4 = 20$
Two operations are done on the variable a to get the expression on the left side of the equation. The variable a is multiplied by 3 and then 4 is subtracted from the product. The operations are undone in reverse order. First addition to undo subtraction, followed by division to reverse multiplication.

$3a - 4 = 20$
$+4 +4$                  Addition property of equality
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$3a = 24$
$\frac{3a}{3} \frac{24}{3}$                 Division property of equality
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$a = 8$                  Solution of the equation
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## Solving multi -step equations

Initial simplifications of expression needs to be done before applying the properties of equalities in solving the equation. The linear equation may also contain variables on either side of the equation. Hence additional steps are required to solve a linear equation in one variable.

Solve the equation:
$4(y+20)=\frac{1}{10}(20y+400)$
$4y+80=\frac{20y}{10}+\frac{400}{10}$                                         Distributive Property
$4y+80 =2y+40$
$-80 -80$                                                Subtraction Property of equality
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$4y = 2y -40$
$-2y =-2y$                                                   Subtraction Property of equality
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$2y = -40$
$\frac{2y}{2}=-\frac{40}{2}$                                                   Division Property of equality
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$y = -20$                                                     Solution of the equation
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