Working with simultaneous equations
Simultaneous equations can be solved algebraically by first eliminating one of the unknowns so the other can be found.
Solving simultaneous equations graphically
Simultaneous equations can be solved algebraically or graphically. Knowledge of plotting linear equations 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 \(y = 2\).
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 \(y = x + 1\).
To draw these graphs, use a table of values:
\[y = x + 1\]
| \[x\] | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|
| \[y\] | 0 | 1 | 2 | 3 | 4 |
\[x + y = 5\]
| \[x\] | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|
| \[y\] | 6 | 5 | 4 | 3 | 2 |
Plot these graphs onto the axes and label each graph.
The point of intersection is (2, 3) which means \(x = 2\) and \(y = 3\).