## The Lesson

Simultaneous equations are a set of several equations with several unknowns. In this lessons, we will be looking at a set of 2 linear equations, where the unknowns are the variables**x**and

**y**:

There are a set of values of these unknowns which solve all the equations at the same time. The solution to the simultaneous equations shown above are:

## Understanding Solving Simultaneous Equations

It is easier to understand solving simultaneous equations with an example.
2x + y = 4 ... (1)
x + 3y = 7 ... (2)

**x**and

**y**are variables. They can take different values. Let's try different pairs of values of

**x**and

**y**.

## x = 2, y = 0

Substitute**x = 2**and

**y = 0**into both equations and see if the left hand side of the equation equals the right hand side of the equation:

Equation (1) ... 2(

*2*) + (*0*) = 4 Equation (1) ... 2 ×*2*+*0*= 4 Equation (1) ... 4 = 4 ? ✔
Equation (2) ...

*2*+ 3(*0*) = 7 Equation (1) ...*2*+ 3 ×*0*= 7 Equation (1) ... 2 = 7 ? ✖**x = 2**and

**y = 0**solves

*Equation (1)*but does not solve

*Equation (2)*. It does not solve both equations at the same time. It is not a solution to the simultaneous equations.

## x = 4, y = 1

Substitute**x = 4**and

**y = 1**into both equations and see if the left hand side of the equation equals the right hand side of the equation:

Equation (1) ... 2(

*4*) + (*1*) = 3 Equation (1) ... 2 ×*4*+*1*= 3 Equation (1) ... 9 = 3 ? ✖
Equation (2) ...

*4*+ 3(*1*) = 7 Equation (1) ...*4*+ 3 ×*1*= 7 Equation (1) ... 7 = 7 ? ✔**x = 4**and

**y = 1**solves

*Equation (1)*but does not solve

*Equation (2)*. It does not solve both equations at the same time. It is not a solution to the simultaneous equations.

## x = 1, y = 2

Substitute**x = 1**and

**y = 2**into both equations and see if the left hand side of the equation equals the right hand side of the equation:

Equation (1) ... 2(

*1*) + (*2*) = 4 Equation (1) ... 2 ×*1*+*2*= 4 Equation (1) ... 4 = 4 ? ✔
Equation (2) ...

*1*+ 3(*2*) = 7 Equation (1) ...*1*+ 3 ×*2*= 7 Equation (1) ... 7 = 7 ? ✔**x = 1**and

**y = 2**solves

*Equation (1)*and

*Equation (2)*. It solves both equations at the same time. It is a solution to the simultaneous equations.

**x = 1**and

**y = 2**solve the simultaneous equations.

## How to Solve Simultaneous Equations

There are 3 ways to solve simultaneous equations.## Elimination

Simultaneous equation can be solved by*eliminating*one of the unknowns by adding or subtracting linear combinations of the equations.

Read more about solving simultaneous equations using elimination

## Substitution

Simultaneous equation can be solved by*substituting*one of the equations into the other.

Read more about solving simultaneous equations using substitution

## Graph

Simultaneous equation can be solved by plotting each equation on a*graph*and seeing where they meet.

Read more about solving simultaneous equations using a graph

## What's in a Name?

Simultaneous equations are so called because the solution solves all the equations at the same time, or simultaneously.## Number of Unknowns = Number of Equations

In the simultaneous equations in this lesson:-
there have been
**2 equations**. -
there have also been
**2 unknowns**(**x**and**y**).

**3**unknowns (e.g.

**x**,

**y**and

**z**), we would need

**3**equations to solve them.