Lesson 1: Solving Linear Systems by Graphing

Module 7: Systems of Linear Equations

Explore

Study the following two photos. Which one is more interesting to you? Which one do you think conforms to the principles of photography? One of the principles of photography is called the rule of thirds. This rule helps a photographer to compose a picture by aligning the key features of the photo to a 3 × 3 grid. You do this by first dividing the photo into thirds both horizontally and vertically. To make your photo interesting, place the key features of the photo at the points of intersection.

 

 

Try this exercise: Print the photos. Use a pencil and a ruler to divide each photo into thirds, along both the width and along the length. Draw grid lines across and down each photo. Do you notice that the lines across the second photo intersect along the face of the man to focus your attention there?

In this lesson you will learn a method for solving linear systems that involves graphing. Just like the rule of thirds in photography, the key to finding correct solutions is found at the points of intersection.

Glossary Terms

Add these terms and their definitions to the Glossary Terms section in your notes. You may also want to add examples that demonstrate how each term is applied.

• linear system/system of equations
• point of intersection
• solution to a system

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Try This 8 - 10

Complete the following in your binder.
Try This 8 - 10 Questions
Use the link below to check your answers to Try This 8 - 10.
Possible TT 8-10 Solutions

You may have noticed in the Try This exercise that when the game involves the guessing of two numbers, a minimum of two clues are required. Likewise, when the game involves the guessing of three numbers, a minimum of three clues are required.

Solving equations with multiple unknowns is a similar undertaking. When you are solving an equation with two unknown variables, it is not possible to do so with only one equation.

For example, the equation x + y = 10 indicates that the sum of two numbers is 10. The solution, however, could be any number of different pairs including 1 and 9, 2 and 8, or even –3 and 13. More clues are required in order to pinpoint the correct values. If the second clue is the equation x = 7, then you could use the two equations together to determine that x = 7 and y = 3. In this case, the equations x + y = 10 and x = 7 form a system of equations, or a set of clues, to determine the value of the unknown variables.

In general, the number of equations must be equal to or greater than the number of unknowns to be determined.

Imagine that one day you decide that you want to eat lunch at one of the fast-food restaurants near your home. You have several choices including a pizza parlour, a sandwich shop, and a burger joint. You could be satisfied with any of the three choices. The next day, you and a friend get together to have lunch. You are still open to going to any of the three restaurants that you were considering the day before, but your friend has suggested a local deli, a hot dog stand, or the sandwich shop. Where should you go for lunch in order to satisfy both of you?

In the Discover section you may have found that solving a system of equations is similar to the situation described in the above paragraph. There are many solutions to a linear equation. In other words, there are many combinations of x-values and y-values that will satisfy a particular equation. However, there may be fewer solutions to a system of equations or fewer x-values and y-values that satisfy each equation in a linear system.

You have also learned from the Discover activity that you can find the solution to a system of equations by graphing.
You have previously learned different techniques for graphing linear relations. Ensure you know the three different methods by the end of the lesson.

The three different ways to graph a linear relation are:

1. Develop a table of values.
2. Change the equation into slope intercept form.
3. Algebraically find the x and y intercepts.

Share 1 - 4

Consider the system of equations:

2x + y = 5

x – y = 10

Work with a partner (if possible) to complete Share questions 1 to 4 to solve the system of equations by graphing.
Share 1 - 4 Questions

Use the provided link to check your work.
Possible S1 - 4  Solutions

Work through the next example in your textbook; then make any necessary revisions to the Share questions you just completed. 

Read

 

Go to the textbook you are using to complete this course to find an example of how to solve a linear system by using a graphing method. Work through the example and pay attention to the strategy used to graph the linear equations in the system and how the solution is verified.

Compare the strategies learned in the example to the one developed in the Share activity. Write down what you feel is important to remember about using the strategies shown in the example.

 

                             

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