Solving simultaneous equations graphically

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).

Graph showing plots of y=x+1 & x+y=3

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

Solving linear equations graphically


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.

Graph axes from -10 to 10

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

To draw these graphs, use a table of values:

\[y = x + 1\]


\[x + y = 5\]


Plot these graphs onto the axes and label each graph.

Graph showing plots of y=x+1 & x+y=5

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.


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

\[y = x^2\]


\[y = x + 2\]


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

Graph showing plots of y=x^2 & y=x+2

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

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