Lesson 6

Discover

 

Try This 1

 

Consider this system of equations comprised of two functions:

 

y = x2 − 2x

y = x + 4

1. Construct a table like the following. Leave room for a fourth column as shown. Fill in the y-coordinates corresponding to the given x-coordinates for each of the functions. The first two rows have been completed as an example.
   
   
    

   x	y = x2 − 2x	y = x + 4
   −3	15	1
   −2	8	2
   −1
   0
   1
   2
   3
   4
   5

   
   
   
2. Use the table to create a graph. Plot the coordinates of the two functions. Join the points of each function with a smooth curve or line.
   
   
3. Show where the solutions to the system of equations are on the graph.
   
   
4. Label the last column of your chart “Difference of Equations.” In this column, evaluate the difference between the y-coordinates of the first two equations. For example, the difference in the first row would be recorded (15 − 1) or 14, and the difference in the second row would be (8 − 2) = 6.
   
   
    

   x	y = x2 − 2x	y = x + 4	Difference of Equations
   −3	15	1	14
   −2	8	2	6
   −1
   0
   1
   2
   3
   4
   5

   
   
   
5. Plot the points made from the coordinates of the x-values in the first column and the y-values from the final column. Join the points with a smooth curve. You may wish to use a different colour from the first two functions.

You may want to work together with a partner to complete the Analysis questions.

 

Analysis

6. How could you determine the x-values of the solutions to the system of equations by looking only at the graph of the third function? Circle the x-values of the solutions on the graph.
   
   
7. Write the equation representing the difference of the two equations. How can you use this equation to algebraically determine the solution to the system of equations?
   
   
8. What if you reversed the order of the subtraction and evaluated the other difference? How would your third graph change? Would you still be able to use that graph to determine the solutions to the system?

Save your work in your course folder.

 

Share 1

 

In Try This 1 you worked through steps to find the solution to a system of equations. You subtracted the two equations to find the difference between them and then graphed this difference to find the solutions. You then found the solutions algebraically. Working with a classmate, discuss the following questions based on your Try This 1 observations. Then write a short paragraph that summarizes the answers.

• How did the solutions found using the graph compare to the solutions found algebraically?

• Did the order in which you subtracted the equations have an effect on the solutions?
  
  
• Both graphing the difference and finding the difference algebraically help you to find the x-values of the solutions. What then should be your next step?

Save your work in your course folder.

 

You will investigate algebraic methods of solving systems of equations in the remainder of the lesson. You may want to refer to your Try This 1 investigation as you explore the concepts presented.