Solution by Graphing

A system of equations can be solved by graphing if there are only two unknowns. If the letters used for the unknowns are not x and y, we can replace them for the purpose of graphing, or simply relabel the axes to suit.

Locating a Unique Solution

If both equations are linear, then the graphs of the equations are two straight lines. If the lines intersect then the coordinates of intersection satisfy both equations.

Solving a system by graphing showing that two non-parallel lines intersect

The (x,y) of the intersection is the solution of the system. For example, the point (6,1) lies on y=-2x+13 and lies on the line y=x-5.

    \begin{align*}y&=-2x+13\\[10pt]1&=-2(6)+13\\[10pt]1&=-12+13\quad\checkmark\end{align}

 

    \begin{align*}y&=x-5\\[10pt]1&=(6)-5\quad\checkmark\end{align}

In this case we say the solution is unique. There is only one point of intersection that lies on two non-parallel straight lines.

Determine that there is no solution

If the equations are linear and they have the same slope then the two lines are parallel and they won’t intersect. There is no solution.

Solving a system by graphing showing that two parallel lines don't intersect

For example on the lines y=2x-16 and y=2x-11 are parallel. The slope of both lines is 2. There is no (x,y) that lies on both lines. Therefore there is no (x,y) that satisfies both equations. In this case we state “no solution” to indicate we have arrived at that conclusion. This is a different conclusion than leaving a blank or writing “idk”.

Infinitely many solutions

If the lines are identical we call them coincident. There are infinitely many solutions.

Solving a system by graphing showing that two coincident lines always intersect

For example, the lines y=10-x and y+x=10 are just different arrangements of the same relation. When graphed the lines are identical. In this case every point that lies on y=10-x lies on y+x=10. In this case, there are infinitely many solutions. In this case we state y=10-x or y+x=10 as our solution to the system.

Explain more

Learn this in a different way from Khan Academy: Introduction to systems of equations

Draw graphs to find the common point

In this practice applet there is a unique solution to each pair of equations.

Position the points given on each one of the two lines: first the blue, then the green. Use the intersection to determine the value of x and of y that satisfies both equations.

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