In the last section, you reviewed how to graph lines when you were given the equation in slope-intercept form. In this section, you will graph two intersecting lines at the same time, and then look for patterns with the point of intersection.

Use the interactive below to graph the two linear equations shown. Use your graph to answer the questions that follow.

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Need additional directions?

Use the interactive to answer the questions below.

• What are the coordinates of the point of intersection? What does this tell you about the x-value and y-value of this point?

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  Check Your Answer

  The coordinates are (4, 3), which means that x = 4 and y = 3.
  
  

• Substitute the x-value and y-value into the first equation and simplify the results. What does this tell you about the coordinates of the point?

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  Check Your Answer

  (3) = 2(4) − 5
  3 = 8 − 5
  3 = 3
  Since the same value is obtained on both sides of the equal sign, the point (4, 3) is a solution to the equation y = 2x − 5.

• Substitute the x-value and y-value into the second equation and simplify the results. What does this tell you about the coordinates of the point?

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  Check Your Answer

  (3) = 1 over 2 1 2 (4) + 1
  3 = 2 + 1
  3 = 3
  Since the same value is obtained on both sides of the equal sign, the point (4, 3) is a solution to the equation y = 1 over 2 1 2 x + 1.

• Do the coordinates of the point of intersection give you a solution to both equations? How do you know?

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  Check Your Answer

  Yes. When the x-value and y-value are substituted into the original equations, you get simplified equations that are true in both cases.

• Repeat this process using the interactive for the two equations, y = 0.4x + 3 and y = −x + 6.5. What do the coordinates of the point of intersection of the graphs of the two lines tell you about the solutions?

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  Check Your Answer

  

  (4) = 0.4(2.5) + 3 4 = 1 + 3 4 = 4	(4) = -(2.5) + 6.5 4 = -2.5 + 6.5 4 = 4
  The ordered pair (2.5, 4) is a solution to both equations.

  

  Pause and Reflect

  When you graph two equations at the same time, the two equations are called simultaneous equations. How can you identify from the graph the solution to two simultaneous equations?

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  Check Your Answer

  The point of intersection, which is also called the intersection point, is the solution to the simultaneous equations.

  Practice

  For each of the following pairs of simultaneous equations, graph the two equations. Then, use the graphs to determine the values of x and y that satisfy both equations at the same time.

  1. y = 0.4x
     y = −x + 7

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     Need a hint?

     Graph both equations by using the y-intercept as the starting point and the slope (as a fraction) to locate additional points. Locate the point of intersection.
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     Check Your Answer

     
  2. y = 2 over 3 2 3 x − 4
     y = 2x

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     Need a hint?

     Graph both equations by using the y-intercept as the starting point and the slope (as a fraction) to locate additional points. Locate the point of intersection.
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     Check Your Answer

     
  3. y = x + 9
     y = −3 over 4 3 4 x − 5

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     Need a hint?

     Graph both equations by using the y-intercept as the starting point and the slope (as a fraction) to locate additional points. Locate the point of intersection.
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     Check Your Answer

     (-8, 1)