Simultaneous equations require algebraic skills to find the values of letters within two or more equations. They are called simultaneous equations because the equations are solved at the same time.

Simultaneous equations can be solved algebraically or graphically. Knowledge of plotting linear and quadratic graphs is needed to solve equations graphically.

To find solutions from graphs, look for the point where the two graphs cross one another. This is the solution point. For example, the solution for the graphs \(y = x + 1\) and \(x + y = 3\) is the coordinate point (1, 2).

The solution to these equations is \(x = 1\) and \(y = 2\).

Solving linear equations graphically

Example

Solve the simultaneous equations \(x + y = 5\) and \(y = x + 1\) using graphs.

To solve this question, first construct a set of axes, making sure there is enough room to plot the two graphs.

Now draw the graphs for \(x + y = 5\) and \(y = x + 1\).

Plot these graphs onto the axes and label each graph.

The point of intersection is (2, 3) which means \(x = 2\) and \(y = 3\).

Solving linear and quadratic equations graphically - Higher

Simultaneous equations that contain a quadratic and equation can also be solved graphically. As with solving algebraically, there will usually be two pairs of solutions.

Example

Solve the simultaneous equations \(y = x^2\) and \(y = x + 2\).

\[y = x^2\]

\[x\]

-3

-2

-1

0

1

2

3

\[y\]

9

4

1

0

1

4

9

\[y = x + 2\]

\[x\]

-1

0

1

2

3

\[y\]

1

2

3

4

5

Plot the graphs on the axes and look for the points of intersection.

The two points of intersection are at (2, 4) and (-1, 1) so \(x = 2\) and \(y = 4\), and \(x = -1\) and \(y = 1\).