# Solving Linear Equations

The equations are those where the given expression is equated to a constant or an expression.  An equation will involve one or more variables. The equations are said to be linear if the degree of each term is 1.  Solution to the linear equations of one or more variable is very much useful in solving for any unknown quantity in any complicated real life problems. This is very much useful in finding the total cost, sales and profit of the multiple products produced in a factory.
Let us discuss about the steps to solve linear equations initially.

## Solving for Linear Equations:

### Linear equations of one variable:

The general for of the linear equations of one variable is px + q = 0, where p, q are real numbers and p is not 0.
The linear equation of one variable has only one solution.

### Solving two step linear equations:

Step 1: Add the additive inverse of the constant term on both sides.

px + q + (- q) = 0 +( - q )
px  + 0 = -q
Step 2: Multiply both sides by the multiplicative inverse of the coefficient of x and simplify on both sides..

Here the multiplicative inverse of p is $\frac{1}{p}$ ( px ) = ( -q ) $\frac{1}{p}$
=>       x = $\frac{-q}{p}$

## Steps for solving linear equations:

If the given linear equation is not of the form px + q = 0, then we use the following steps to solve the linear equations:
1. Apply FOIL or PEMDAS and simplify the expressions by combining the like terms on both sides of the equations.
2. Add the additive inverse of the variables on the right side on both sides of the equation so that the variable term is eliminated on the right side of the equation.
3. Add the additive inverse of the constant term on the left side of the equation, on both sides of the equation so that the constant term is eliminated on the left side of the equation.
4. Use the Step 2 of the two step method to find the value of the variable. (i.e), multiply both sides by the multiplicative inverse of the coefficient of x and simplify to get the value of x.

Cross Multiplication: When $\frac{a}{b}$  =  $\frac{c}{d}$ , then a x d = b x c.
We can use cross multiplication method when we have single fraction on either side of the equation.

## Solving linear equations Examples:

#### Solution:

We have 4x - 27 = 5
4x - 27 + 27  = 5 + 27
4x = 32
x = 32/4 = 8

#### Solution:

We have 4 ( x- 3) + 7 = 5 ( 4 - x) + 20
Applying Distributive property,
4x - 12 + 7 = 20 - 5x + 20
=>   4x   - 5 = 40 - 5x
[ combining the like terms on the left
side and right side of the equation]
=>   4x + 5x - 5 = 40 - 5x + 5x [ adding 5x on both sides ]
=>          9x -5  = 40
=     9x - 5 + 5 = 40 + 5 [ adding 5 on both sides ]
=>              9x  = 45
=>          $\frac{1}{9}$ (9x) = $\frac{1}{9}$ (45)
=>                  x = 5

The above examples will definitely help in solving linear equations of one variable.
Using steps to solving linear equations let us practice few problems.

## Practice Questions:

1. Solve : 5x + 27 = -8
2. Solve : $\frac{4x+5}{4}$  = $\frac{4x-7}{5}$
3. Solve : 3 ( 8x - 5) + 20 = 7 ( 3 - x) + 2( x-3).

*****