Mathematics » Introducing Graphs » Graphing Linear Equations

# Recognizing the Relation Between the Solutions of an Equation and its Graph

## Recognizing the Relation Between the Solutions of an Equation and its Graph

In the previous topic, we found a few solutions to the equation $$3x+2y=6$$. They are listed in the table below. So, the ordered pairs $$\left(0,3\right)$$, $$\left(2,0\right)$$, $$\left(1,\frac{3}{2}\right)$$, $$\left(4,-3\right)$$, are some solutions to the equation$$3x+2y=6$$. We can plot these solutions in the rectangular coordinate system as shown on the graph at right. Notice how the points line up perfectly? We connect the points with a straight line to get the graph of the equation $$3x+2y=6$$. Notice the arrows on the ends of each side of the line. These arrows indicate the line continues. Every point on the line is a solution of the equation. Also, every solution of this equation is a point on this line. Points not on the line are not solutions!

Notice that the point whose coordinates are $$\left(-2,6\right)$$ is on the line shown in the figure below. If you substitute $$x=-2$$ and $$y=6$$ into the equation, you find that it is a solution to the equation. So $$\left(4,1\right)$$ is not a solution to the equation $$3x+2y=6$$ . Therefore the point $$\left(4,1\right)$$ is not on the line.

This is an example of the saying,” A picture is worth a thousand words.” The line shows you all the solutions to the equation. Every point on the line is a solution of the equation. And, every solution of this equation is on this line. This line is called the graph of the equation $$3x+2y=6$$.

### Definition: Graph of a Linear Equation

The graph of a linear equation $$Ax+By=C$$ is a straight line.

• Every point on the line is a solution of the equation.
• Every solution of this equation is a point on this line.

## Example

The graph of $$y=2x-3$$ is shown below. For each ordered pair decide

1. Is the ordered pair a solution to the equation?
2. Is the point on the line?
1. $$\left(0,3\right)$$
2. $$\left(3,-3\right)$$
3. $$\left(2,-3\right)$$
4. $$\left(-1,-5\right)$$

Substitute the $$x$$- and $$y$$-values into the equation to check if the ordered pair is a solution to the equation. Plot the points A: $$\left(0,-3\right)$$ B: $$\left(3,3\right)$$ C: $$\left(2,-3\right)$$ and D: $$\left(-1,-5\right)$$.

The points $$\left(0,-3\right)$$, $$\left(3,3\right)$$, and $$\left(-1,-5\right)$$ are on the line $$y=2x-3$$, and the point $$\left(2,-3\right)$$ is not on the line. The points which are solutions to $$y=2x-3$$ are on the line, but the point which is not a solution is not on the line.