**Simultaneous Equation:**

In mathematics, an arrangement of concurrent conditions, otherwise called an arrangement of conditions, is a limited arrangement of conditions for which regular arrangements are looked for. A condition framework is generally characterized in an indistinguishable way from single conditions in particular as a:

- System of linear equations
- System of bilinear equations
- System of polynomial equations
- System of ordinary differential equations
- System of partial differential equations
- System of difference equations

Simultaneous equations are those two equations having two unknown variables. They are called simultaneous because they must both be solved at the same time. The first step is to try to eliminate one of the unknown variables.

**Simultaneous Equation Example:**

Example

Solve these simultaneous equations and find the values of x and y.

Equation 1: 2x + y = 7

Equation 2: 3x - y = 8

Add the two equations to eliminate the ys:

2x + y = 7

3x - y = 8

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5x = 15

x = 3

Now you can put x = 3 in either of the equations.

Substitute x = 3 into the equation 2x + y = 7:

6 + y = 7

y = 1

So the answers are x = 3 and y = 1

Example

*Solve the simultaneous equations:*

Equation 1: y - 2x = 1

Equation 2: 2y - 3x = 5

Rearranging Equation 1, we get y = 1 + 2x

We can replace the 'y' in equation 2 by substituting it with 1 + 2x

Equation 2 becomes: 2(1 + 2x) - 3x = 5

2 + 4x - 3x = 5

2 + x = 5

x = 3

Substituting x = 3 into Equation 1 gives us y - 6 = 1, so y = 7.

Example

Solve these simultaneous equations by drawing graphs:

2x + 3y = 6

4x - 6y = - 4

For example, to draw the line 2x + 3y = 6 pick two easy numbers to plot. One when x = 0 and one where y= 0

When x = 0 in the equation 2x + 3y = 6

This means 3y = 6 so y = 2

So one point on the line is (0, 2)

When y = 0

2x = 6 so x = 3

So another point on the line is (3 ,0)

In an exam, only use this method if you are prompted to by a question. It is usually quicker to use algebra if you are not asked to use graphs.

**Simultaneous Equation Worksheet:**

Solve the following equations for the value of x and y.

- 4x + 2y = 24

3x + 3y = 24

- 5x + 5y = 30

4x - 5y = 33

- 4x - 4y = 40

2x + y = 17

- 2x - 3y = 15

x - 3y = 12

- 2x - 3y = 7

5x + 3y = 7

- 3x - 2y = 28

2x - 4y = 24

- 3x + 5y = 48

2x + 5y = 42

- x + y = 3

2x - 4y = 12

- 3x - 2y = 5

2x + y = 22

- 5x + 4y = 29

3x - y = 14