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.

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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.

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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.

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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.

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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.

 

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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.

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