## Simultaneous Equations: Substitution Method

This is exactly the same method we use to find where two lines intersect.

Process:

- Rearrange one of the equations to make one of the variables a subject
- Substitute for this variable into the other equation
- Solve the resulting equation
- Substitute these solutions into one of the original equations to find the values of the other variable

Example

Solve the system of equations

Substitute eqn 1 into eqn 2

Substituting into eqn 1 gives

Solve the system of equations

Rearrange eqn 2 to make y the subject

Substitute eqn 3 into eqn 1

Factorising gives

Substituting these values into eqn 3 gives

So the solutions are

Solve the system of equations

Rearranging eqn 2 to make y the subject gives

Substituting eqn 3 into eqn 1 gives

Substituting the value of x into eqn 3 gives

## Simultaneous equations: Elimination method

Simultaneous equations are sytems of equations where there are a number of variables.

Our task is to find the values of the variables that satisfy all the equations.

For this method we will solve equations of the type

With these equations we can do one of three things

- Multiply by a scaler.
- Add the equations together
- Subtract one equation from another

Multiplying by a scaler

eqn 1 multiplied by 6 gives

Adding two equations

eqn 1 + eqn 2 gives

Subtract one equation from another

eqn 1 – eqn 2 gives

### Type 1 Simultaneous equations

With this type of problem the y terms are the same size but different signs.

To solve these we:

- Add the equations together (this will cancel out the y terms)
- Solve the resulting equation to find the value of x
- Substitute the x value into one of the original equations
- Solve the resulting equation to find the value of y
- Check by substituting x and y into the other equation

Examples

Solve

This is a type 1 problem because we have two 4y’s, but one is positive and the other is negative.

So eqn 1 + eqn 2 gives

Substitute into eqn 1

gives

Check in eqn 2

true

Solve

This is a type 1 problem because we have two 5y’s, but one is positive and the other is negative.

So eqn 1 + eqn 2 gives

Substitute into eqn 2 (it is easier to deal with positives)

gives

Check in eqn 1

true

### Type 2 Simultaneous equations

With this type of problem the y terms are the same size and sign.

To solve these we:

- Subtract one equation from the other (this will cancel out the y terms)
- Solve the resulting equation to find the value of x
- Substitute the x value into one of the original equations
- Solve the resulting equation to find the value of y
- Check by substituting x and y into the other equation

Examples

Solve

This is a type 2 problem because we have two +2y’s.

So eqn 2 – eqn 1 gives

Substitute into eqn 1

gives

Check in eqn 2

true

Solve

This is a type 2 problem because we have two -4y’s.

So eqn 2 – eqn 1 gives

Substitute into eqn 1

gives

Check in eqn 2

true

### Type 3 Simultaneous equations

With this type of problem the y terms are different.

To solve these we:

- Multiply each equation by the other equations positive y coefficient
- Solve the resulting type 1 or type 2 problem

Examples

Solve

Multiply eqn 1 by 2 (that is the number of y’s in eqn 2)

Multiply eqn 2 by 4 (that is the number of y’s in eqn 1)

Equations 3 and 4 now make a type 2 problem

Subtracting eqn 3 from eqn 4 gives

Substitute into eqn 2

Check in eqn 1

true

Solve

Multiply eqn 1 by 3 (that is the number of y’s in eqn 2)

Multiply eqn 2 by 2 (that is the number of y’s in eqn 1)

Equations 3 and 4 now make a type 1 problem

Adding eqn 3 and eqn 4 gives

Substitute into eqn 2

Check in eqn 1

true