## 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

- Is the ordered pair a solution to the equation?
- Is the point on the line?

- \(\left(0,3\right)\)
- \(\left(3,-3\right)\)
- \(\left(2,-3\right)\)
- \(\left(-1,-5\right)\)

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.