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Recognizing the Relationship Between the Solutions of An Equation and Its Graph

Recognizing the Relationship Between the Solutions of An Equation and Its Graph

In the previous section, we found several solutions to the equation \(3x+2y=6\). They are listed in the table below. So, the ordered pairs \(\left(0,3\right)\), \(\left(2,0\right)\), and \(\left(1,\frac{3}{2}\right)\) are some solutions to the equation \(3x+2y=6\). We can plot these solutions in the rectangular coordinate system as shown in the figure below.

\(3x+2y=6\)
\(x\) \(y\) \(\left(x,y\right)\)
0 3 \(\left(0,3\right)\)
2 0 \(\left(2,0\right)\)
1 \(\frac{3}{2}\) \(\left(1,\frac{3}{2}\right)\)

The figure shows four points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the four points at (0, 3), (1, three halves), (2, 0), and (4, negative 3). The four points appear to line up along a straight line.

Notice how the points line up perfectly? We connect the points with a line to get the graph of the equation \(3x+2y=6\). See the figure below. Notice the arrows on the ends of each side of the line. These arrows indicate the line continues.

The figure shows a straight line drawn through four points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the four points at (0, 3), (1, three halves), (2, 0), and (4, negative 3). A straight line with a negative slope goes through all four points. The line has arrows on both ends pointing to the edge of the figure. The line is labeled with the equation 3x plus 2y equals 6.

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.

The figure shows a straight line and two points and on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the two points and are labeled by the coordinates “(negative 2, 6)” and “(4, 1)”. The straight line goes through the point (negative 2, 6) but does not go through the point (4, 1).

The figure shows a series of equations to check if the ordered pair (negative 2, 6) is a solution to the equation 3x plus 2y equals 6. The first line states “Test (negative 2, 6)”. The negative 2 is colored blue and the 6 is colored red. The second line states the two- variable equation 3x plus 2y equals 6. The third line shows the ordered pair substituted into the two- variable equation resulting in 3(negative 2) plus 2(6) equals 6 where the negative 2 is colored blue to show it is the first component in the ordered pair and the 6 is red to show it is the second component in the ordered pair. The fourth line is the simplified equation negative 6 plus 12 equals 6. The fifth line is the further simplified equation 6equals6. A check mark is written next to the last equation to indicate it is a true statement and show that (negative 2, 6) is a solution to the equation 3x plus 2y equals 6.

So the point \(\left(-2,6\right)\) is a solution to the equation \(3x+2y=6\). (The phrase “the point whose coordinates are \(\left(-2,6\right)\)” is often shortened to “the point \(\left(-2,6\right)\).”)

The figure shows a series of equations to check if the ordered pair (4, 1) is a solution to the equation 3x plus 2y equals 6. The first line states “What about (4, 1)?”. The 4 is colored blue and the 1 is colored red. The second line states the two- variable equation 3x plus 2y equals 6. The third line shows the ordered pair substituted into the two- variable equation resulting in 3(4) plus 2(1) equals 6 where the 4 is colored blue to show it is the first component in the ordered pair and the 1 is red to show it is the second component in the ordered pair. The fourth line is the simplified equation 12 plus 2 equals 6. A question mark is placed above the equals sign to indicate that it is not known if the equation is true or false. The fifth line is the further simplified statement 14 not equal to 6. A “not equals” sign is written between the two numbers and looks like an equals sign with a forward slash through it.

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. See the figure below, already displayed above.

The figure shows a straight line drawn through four points on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the four points at (0, 3), (1, three halves), (2, 0), and (4, negative 3). A straight line with a negative slope goes through all four points. The line has arrows on both ends pointing to the edge of the figure. The line is labeled with the equation 3x plus 2y equals 6.

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 \(3x+2y=6\).

Graph of a Linear Equation

The graph of a linear equation\(Ax+By=C\) is a 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.

The figure shows a straight line on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. The straight line has a positive slope and goes through the y-axis at the (0, negative 3). The line is labeled with the equation y equals 2x negative 3.

For each ordered pair, decide:

Is the ordered pair a solution to the equation?

Is the point on the line?

A \(\left(0,-3\right)\) B \(\left(3,3\right)\) C \(\left(2,-3\right)\) D \(\left(-1,-5\right)\)

Solution

Substitute the x- and y- values into the equation to check if the ordered pair is a solution to the equation.

  1. The figure shows a series of equations to check if the ordered pairs (0, negative 3), (3, 3), (2, negative 3), and (negative 1, negative 5) are a solutions to the equation y equals 2x negative 3. The first line states the ordered pairs with the labels A: (0, negative 3), B: (3, 3), C: (2, negative 3), and D: (negative 1, negative 5). The first components are colored blue and the second components are colored red. The second line states the two- variable equation y equals 2x minus 3. The third line shows the four ordered pairs substituted into the two- variable equation resulting in four equations. The first equation is negative 3 equals 2(0) minus 3 where the 0 is colored clue and the negative 3 on the left side of the equation is colored red. The second equation is 3 equals 2(3) minus 3 where the 3 in parentheses is colored clue and the 3 on the left side of the equation is colored red. The third equation is negative 3 equals 2(2) minus 3 where the 2 in parentheses is colored clue and the negative 3 on the left side of the equation is colored red. The fourth equation is negative 5 equals 2(negative 1) minus 3 where the negative 1 is colored clue and the negative 5 is colored red. Question marks are placed above all the equal signs to indicate that it is not known if the equations are true or false. The fourth line shows the simplified versions of the four equations. The first is negative 3 equals negative 3 with a check mark indicating (0, negative 3) is a solution. The second is 3 equals 3 with a check mark indicating (3, 3) is a solution. The third is negative 3 not equals 1 indicating (2, negative 3) is not a solution. The fourth is negative 5 equals negative 5 with a check mark indicating (negative 1, negative 5) is a solution.
  2. 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 figure shows a straight line and four points and on the x y-coordinate plane. The x-axis of the plane runs from negative 7 to 7. The y-axis of the plane runs from negative 7 to 7. Dots mark off the two points and are labeled by the coordinates (negative 1, negative 5), (0, negative 3), (2, negative 3), and (3, 3). The straight line, labeled with the equation y equals 2x negative 3 goes through the three points (negative 1, negative 5), (0, negative 3), and (3, 3) but does not go through the point (2, negative 3).

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 that are solutions to \(y=2x-3\) are on the line, but the point that is not a solution is not on the line.

This lesson is part of:

Graphs and Equations

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