# Simultaneous Equations

## Definition

Simultaneous equations includes a set of few independent equations. These equations contains two or more variables. To solve the equations, we need to find the values of the variables included in these equations. These equations can be solved using various methods.

Simultaneous equation has a general form which is written as

 ax +by = c dx + ey = f

## Simultaneous Equation Example

Let us now understand how to solve simultaneous equations through the above mentioned methods Simultaneous and Elimination method. We will get the value of a and b to find the solution for the same. X and y are the two variables in these equations.

### Simultaneous Equations Elimination Method

Solve the two pairs of simultaneous equations by elimination method.

1. Example: 4a + 5b = 12,

3a – 5b = 9

Solution: The two given equations are

4a + 5b = 12 …….(1)

3a – 5b = 9……….(2)

Let’s learn the steps for the same:

1. The coefficient of variable ’b’ is equal and has the opposite sign to the other equation. Add the equations 1 and 2 to eliminate the variable ‘b’.
2. The like terms will be added.

(4a+3a) +(5b – 5b) = 12 + 9

7a = 21

3. Divide the equation in step 2, i.e 7a = 21 on both sides by 7

7a/7 = 21/7

4. We will get a = 3
5. Now, put the value of a in equation (1)

4(3) + 5b = 12,

12 + 5b = 12

5b = 12/12

b = 0/5 = 0

6. So, the solution for these equations are a =3 and b = 0.

### Simultaneous Equations Substitution

Solve the two pairs of simultaneous equations by elimination method.

2. Example: b= a + 2

a + b = 4.

Soln: The two given equations are

b = a + 2 ————–(1)

a + b = 4 ————–(2)

We will solve it stepwise:

1. Put the value of b into the second equation. We will get,

a + (a + 2) = 4

1. Solve for a

a +a + 2 = 4

2a + 2 = 4

2a = 4 – 2

a = 2/2 = 1

1. Substitute this value of a in equation 1

b = a + 2

b = 1 + 2

b = 3

1. So the values of variables a and b are 1 and 3 respectively.