**Syllabus Requirement**

Students need to learn to solve

**Simultaneous equations by substitution**, including examples where

**one equation is linear and one equation is quadratic**.

Simultaneous equations are sets of equations with multiple unknowns that have the same value in each equation.

For example, the two equations below together form a set of simultaneous equations:
$$x + y = 3$$
$$x - y = -1$$
The primary method of solving simultaneous equations is through **substitution** although sometimes the method of **elimination** can be quicker.

Both are explained below:

**Substitution**

- Choose one of the equations and rearrange to get one of the unknowns (usually ??x?? or ??y??) by itself on one side of the equation.
- Now in the other equation, substitute in for the unknown that you have rearranged for in step 1.
- This should result in a value for the second unknown, which can be substituted back into the first equation to get a value for the remaining unknown.

**Elimination**

- Choose one of the unknowns, the goal is to eliminate this unknown from both equations.
- To eliminate the unknown you will need to either add both equations, or subtract one from the other.
- It may be necessary to multiply one of the equations by a constant to ensure that the unknown variable is eliminated.
- Once eliminated, you should have an equation that is solvable for the other unknown.
- Once you have solved for one of the unknowns, substitute that value back into one of the equations to get the value of the other unknown.

We'll now run through an example of each method:

**Substitution**

**Example 1**

**Question**

Solve for ??x?? and ??y?? the following two simultaneous equations:

- $$2xy + y = 10$$
- $$x + y = 4$$

Show Solution

**Elimination**

**Example 2**

**Question**

Solve the below simultaneous equations using the elimination method:

- $$ 3x + y = 9$$
- $$ 2x - y = 1$$

Show Solution

The table below contains every exam question that has been asked on this topic, this includes normal papers from both January and June sittings, International papers and Specimen papers.

To see an exam question and solution simply click on the load question icon (), the question will appear below the table and the solution can be shown by clicking the "Show Solution" button that also appears.

The full exam paper and mark schemes are also available for download by clicking on the download icons in each row of the table ().

- Exam questions most often will simply ask you to solve two simultaneous equations. Typically, you will be free to choose whether to use substitution or elimination as a method (although you should be familiar with both - questions have sometimes asked for a particular method to be used).
- Unless specified otherwise, you should solve simultaneous equations algebraically and not graphically.
- Often questions ask that the student finds the point where two lines meet. This will simply involve solving the two simultaneous equations algebraically again (unless it is specified to be done graphically). Remember, that once you have solved for either the ??x?? or ??y?? coordinate you will need to calculate the other coordinate as well if the question specifies that you find the point where the two lines or curves cross.