Mathematics » Equations and Inequalities » Solving Simultaneous Equations

Solving Graphically

Solving Graphically

Simultaneous equations can also be solved graphically. If the graphs of each linear equation are drawn, then the solution to the system of simultaneous equations is the coordinates of the point at which the two graphs intersect.

For example:

\begin{align*} x & = 2y \qquad \ldots (1) \\ y & = 2x – 3 \qquad \ldots (2) \end{align*}

The graphs of the two equations are shown below.

4e6a26f9cb3dae7bb899566dda1f06fe.png

The intersection of the two graphs is \((2;1)\). So the solution to the system of simultaneous equations is \(x=2\) and \(y=1\). We can also check the solution using algebraic methods.

Substitute equation \((1)\) into \((2)\):

\begin{align*} x & = 2y \\ \therefore y & = 2(2y) – 3 \end{align*}

Then solve for \(y\):

\begin{align*} y – 4y & = -3 \\ -3y & = -3 \\ \therefore y & = 1 \end{align*}

Substitute the value of \(y\) back into equation \((1)\):

\begin{align*} x & = 2(1) \\ \therefore x & = 2 \end{align*}

Notice that both methods give the same solution.

You can use an online tool such as graphsketch to draw the graphs and check your solution.

Example

Question

Solve the following system of simultaneous equations graphically:

\begin{align*} 4y + 3x & = 100 \qquad \ldots (1) \\ 4y – 19x & = 12 \qquad \ldots (2) \end{align*}

Write both equations in form \(y=mx + c\)

\begin{align*} 4y + 3x & = 100 \\ 4y & = 100 – 3x \\ y & = -\cfrac{3}{4}x + 25 \end{align*}\begin{align*} 4y – 19x & = 12 \\ 4y & = 19x + 12 \\ y & = \cfrac{19}{4}x + 3 \end{align*}

Sketch the graphs on the same set of axes

87973ba835ca238d1caa651d5e178a71.png

Find the coordinates of the point of intersection

The two graphs intersect at \((4;22)\)

Write the final answer

\begin{align*} x & = 4 \\ y & = 22 \end{align*}

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