Solve systems of linear equations by graphing

A system of two linear equations represents two lines in the coordinate plane, and its solution is the ordered pair that lies on both lines—their point of intersection. Graphing develops interpretation of the possible cases: one intersection gives one solution, parallel distinct lines give no solution, and coincident lines give infinitely many solutions; coordinates read from a graph may be approximate and can be checked in both equations. The scope is two-variable linear systems, not nonlinear or higher-dimensional systems.

Detailed Explanation: Solve systems of linear equations by graphing

A system of linear equations is a pair of equations with the same variables. Its solution is the ordered pair $(x,y)$ that makes both equations true. On a graph, this solution is the point where the two lines intersect.

Example

Solve by graphing:

\begin{cases} y=2x+1\\ y=-x+4 \end{cases}

1. Graph the first line.

For $y=2x+1$ :

The $y$ -intercept is $1$ , so plot $(0,1)$ .
The slope is $2=\frac{2}{1}$ . From $(0,1)$ , go up $2$ and right $1$ to plot $(1,3)$ .

Draw a line through these points.

2. Graph the second line.

For $y=-x+4$ :

The $y$ -intercept is $4$ , so plot $(0,4)$ .
The slope is $-1=\frac{-1}{1}$ . From $(0,4)$ , go down $1$ and right $1$ to plot $(1,3)$ .

Draw a line through these points.

3. Find the intersection.

The lines intersect at $(1,3)$ , so the solution is

\boxed{(1,3)}

4. Check the solution.

Substitute $x=1$ and $y=3$ into both equations:

3=2(1)+1

and

3=-(1)+4

Both equations are true, so $(1,3)$ is the solution.

Two lines can also have no intersection if they are parallel, giving no solution. If the lines overlap completely, they have infinitely many solutions.

Learn by doing: Solve systems of linear equations by graphing

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Types of Solutions - Graph to Zero, One, or Infinite

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