Lesson 6

Lesson 6: Solving Systems of Equations Algebraically

 

Focus

 

The World’s Fair of 1893 was an opportunity for the city of Chicago, Illinois, to showcase its architecture, technology, and the spirit of America. When the architects and designers were charged with the task of bringing the vision of a world’s fair to life, they were faced with the challenge of outdoing the World Exhibition of 1889, held in Paris to commemorate the centennial of the French Revolution. The Eiffel Tower was the centrepiece of the World Exhibition. Standing at 300 m, it became the undisputed tallest structure in the world, dwarfing the Washington Monument by twice the size.

 

In order to “out-Eiffel Eiffel,” and also to attract as many visitors as possible, the architects designed a lavish city within a city with magnificent buildings, exquisitely manicured landscapes, and the world’s first Ferris wheel. While the Ferris wheel did not rival the Eiffel tower in terms of height, reaching only 80.4 m (or approximately 27 storeys), the Ferris wheel was commanding in its own right. With a total of 36 cars, each of which could accommodate 60 people, the Ferris wheel could handle 2160 passengers!

 

The design and construction of an attractive and entertaining fairground depends to a great extent on mathematics. Mathematics is used in the design of rides, such as a rollercoaster or a Ferris wheel, and in impressive shows, such as a choreographed water show or fireworks.

 

In this lesson you will continue to investigate linear-quadratic and quadratic-quadratic systems. You will learn how to solve these types of systems algebraically. You will use methods that are similar to those that you have previously studied.

 

Outcomes

 

At the end of this lesson you will be able to

• determine and verify the solution of a system of linear-quadratic or quadratic-quadratic equations algebraically
  
  
• solve a problem that involves a system of linear-quadratic or quadratic-quadratic equations and explain the strategy used
  
  
• model a situation using a system of linear-quadratic or quadratic-quadratic equations

Lesson Questions

 

You will investigate the following questions:

• Why do systems of equations yield the correct solutions even after they have been manipulated algebraically?
  
  
• Why is it important to be able to solve systems of equations algebraically as well as graphically?

Assessment

Your assessment may be based on a combination of the following tasks:

• completion of the Lesson 6 Assignment (Download the Lesson 6 Assignment and save it in your course folder now.)
  
  
• course folder submissions from Try This and Share activities
  
  
• additions to Module 4 Glossary Terms and Formula Sheet
  
  
• work under Project Connection

Launch

 

Do you have the background knowledge and skills you need to complete this lesson successfully? This section, which includes Are You Ready? and Refresher, will help you find out.

 

Before beginning this lesson you should be able to

• solve systems of linear equations by substitution and by elimination
  
  
• apply the Pythagorean theorem
  
  

Are You Ready?

 

Complete these questions. If you experience difficulty and need help, visit Refresher or contact your teacher.

1. Solve the following systems of equations by substitution.
   
   
   1. x + y = 11
      y = 2x + 5
      
      Answer
      
      
   2. 3a + 4b = 1
      a + 2b = 7
      
      Answer
      
      
2. Solve the following systems of equations by elimination.
   
   
   1. 2x + 5y = 18
      4x − 3y = 10
      
      Answer
      
      
   2. 9p − 8q = 126
      7p + 17q = −42
      
      Answer
      
      
3. State the Pythagorean theorem. Answer
   
   
4. Use the Pythagorean theorem to determine the length of the hypotenuse of the triangle shown. Assume that each grid square is equal to one unit. Answer
   
   
    
   
   

 

Refresher

 

Use the following to review the topics you experienced difficulties with in the Are You Ready? section.

 

Use the Solving Linear Systems Algebraically series of interactive lessons. Complete the following lessons:

• Solving Systems by Substitution
  
  
• Solving Systems by Elimination (Addition or Subtraction)

You may also want to try the last two lessons, which deal with applications and strategy selection.

 

 

 

 

---

Use Pythagorean Theorem to review the definition of this important theorem in mathematics. The demonstration applet gives a visual representation of the theorem.

 

 

 

 

Go back to the Are You Ready? section and try the questions again. If you are still having difficulty, contact your teacher.

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.

Explore

 

In an animated fountain, water streams reach different heights and may also take on different colours. A musical fountain is an extension of this concept: the movement of the streams is synchronized with upbeat music. The Fountains of Bellagio in Las Vegas, Nevada, have the power to shoot streams of water up to 24 storeys high, and, at any given point during a show, there could be up to 65 000 L of water in the air! The water show has been choreographed to music by a variety of different artists, such as Elvis Presley, Céline Dion, the Mormon Tabernacle Choir, and Luciano Pavarotti.

 

In this lesson you will solve systems of linear-quadratic and quadratic-quadratic equations algebraically. Later in the lesson you will model a situation related to a musical fountain using a system of equations.

 

Solving Linear Equations Algebraically

 

Recall from your previous studies two algebraic methods used to solve a system of linear equations.

 

The following chart summarizes the main idea of each method.

 

Substitution	Elimination
Rearrange one of the equations for a single variable. Substitute the expression for the variable into the other equation.	Multiply the equations by constants such that the coefficients of the x-terms or the y-terms either match or have a sum of zero.

 

Try This 2

 

Consider the following linear-quadratic system.

 

y = −x2 + 3

x + 2y = 6

1. Solve the system by substitution.
   
   
2. Solve the system by elimination.
   
   
3. Verify your solutions. You may choose to do this by substituting the determined values into the original equations or by graphing.

Save your work in your course folder.

 

Share 2

 

There are four main ways of solving the system of equations from Try This 2—two variations each of substitution and elimination.

1. Compare your solutions from Try This 2 with a partner. Between the two of you, complete a table like the one shown by copying the appropriate solutions into the appropriate columns.
   
   
    

   Substitution: Isolate y	Substitution: Isolate x	Elimination: Eliminate y	Elimination: Eliminate x

   
   
2. For any solutions that are not covered by either your work or your partner’s work, consider together how you would solve the system by the method indicated. Then record your solutions in the appropriate columns.
   
   
3. Discuss which method(s) you prefer and give reasons why. Under what circumstances would you use some of the other methods that you did not prefer for the system in Try This 2?

Save your work in your course folder.

 

Lesson 6 Summary

 

In this lesson you investigated the following questions:

• Why do systems of equations yield the correct solutions even after they have been manipulated algebraically?

• Why is it important to be able to solve systems of equations algebraically as well as graphically?

You previously learned how to solve systems of linear-quadratic and quadratic-quadratic equations based on graphical methods. In this lesson you studied how to solve these systems using algebraic techniques. You used substitution and elimination strategies to manipulate the equations in each system. You learned that you can algebraically manipulate the equations in a system without changing the solutions. You verified this conclusion by graphing, as shown in the following graph from the Discover section of the lesson.

 

 

Knowing how to algebraically solve math problems, such as systems of equations, is an important skill for understanding math principles. While a graphical solution is often easy to obtain, such a solution process does not reveal the process by which the solution is obtained from a mathematical standpoint. Algebraic methods, when applied correctly, can take you step-by-step from problem to solution, which leads, in turn, to understanding and insight.

 

In the remaining lessons of this module, you will consider linear and quadratic inequalities, studying them from both a graphical and algebraic perspective.