Section 3.3: Solving Systems of Linear Equations in Two Variables Graphically and Numerically

Section 3.3:

Solving Systems of Linear Equations in Two

Variables Graphically and Numerically

3.3 Lecture Guide: Solving Systems of Linear Equations in Two Variables Graphically and Numerically

Objective: Solve a system of linear equations using graphs and tables.

1. Determine which of the following points are solutions of the equation

(a)

32

4y x

4, 1

1. Determine which of the following points are solutions of the equation

(b)

32

4y x

2, 3.5

1. Determine which of the following points are solutions of the equation

(c)

32

4y x

3, 1

1. Determine which of the following points are solutions of the equation

(d)

32

4y x

8,8

2. Determine whether the ordered pair 2, 2 is a solution

of the system of linear equations

3 2

4

y x

y x

3. Determine whether the ordered pair 1, 2 is a solution

of the system of linear equations

2 4

3 1

x y

x y

4. Consider the graph of the system: 243

xy

y x

-5

5

-5 5

y

x

(a) Determine the point of intersection.

(b) Verify that this point satisfies both equations.

5. Give the solution of each system of linear equations.

(a)

Solution: __________________

5. Give the solution of each system of linear equations.

(b)

1,4,1 by 5,1,1

Solution: __________________

6. Solve each equation for y. Then use your calculator to determine the solution of the system of linear equations.

5

3 3

x y

x y

Solution: __________________

7. Use your calculator to determine the rational solution of the system of linear equations. See Calculator Perspective 3.3.1 for help.

3 5

4 8

y x

y x

Solution: __________________

Objective: Identify inconsistent systems and systems of dependent linear equations.

Recall that we have three possibilities when solving linear equations in one variable: a ________________ equation, an __________________, or a _____________________.

8. Solve each equation and identify its type from the list above.

2 3x x (a)

8. Solve each equation and identify its type from the list above.

(b) 2 2x x (c) 2 1 1x x

A solution of a linear equation in one variable is an _____-value that results in both sides of the equation having the _____________________ value.

9. Create a table of values using 1y = the left side and

2y = the right side of each equation in problem 8.

1 2

3

2

1

0

1

2

3

x y y

2 3x x (a)

9. Create a table of values using 1y = the left side and

2y = the right side of each equation in problem 8.

(b) 1 2

3

2

1

0

1

2

3

x y y

2 2x x

9. Create a table of values using 1y = the left side and

2y = the right side of each equation in problem 8.

1 2

3

2

1

0

1

2

3

x y y

(c) 2 1 1x x

In a table of values, a solution of a linear equation in one variable is an _____-value at which the two _____-values are _____________________.

10. Sketch the graphs of 1y = the left side and 2y of each equation in problem 8.

= the right side

-5

5

-5 5

y

x

(a) 2 3x x

10. Sketch the graphs of 1y = the left side and 2y = the right side

-5

5

-5 5

y

x

(b) 2 2x x

of each equation in problem 8.

10. Sketch the graphs of 1y = the left side and 2y = the right side

-5

5

-5 5

y

x

(c) 2 1 1x x

of each equation in problem 8.

On a graph, a solution of a linear equation in one variable is an _____-value at which the two graphs _____________________.

A solution of a system of equations is an ____________ __________, ,x y , that satisfies each equation in the system.

11. Problems 8, 9, and 10 can be used to explore the three cases for linear equations in one variable and the three cases for solutions of systems of two linear equations in two variables. Use problems 8, 9, and 10 to classify each system as one of the following:

A. A consistent system of independent linear equations having exactly one solution.

B. An inconsistent system of linear equations having no solution.

C. A consistent system of dependent linear equations having an infinite number of solutions.

2

3

y x

y x

(a)2

2

y x

y x

(b)2 1

1

y x

y x

(c)

Use the slope-intercept form of each line to determine the number of solutions of the system and then classify each system as a consistent system of independent equations, an inconsistent system, or a consistent system of dependent equations. Then solve each system using a graph or a table.

12. 24

32

43

y x

y x

Consistent or inconsistent system?

Dependent or independent equations?

Solution: ____________________

Consistent or inconsistent system?

Dependent or independent equations?

Solution: ____________________

37

43

24

y x

y x

13.

Use the slope-intercept form of each line to determine the number of solutions of the system and then classify each system as a consistent system of independent equations, an inconsistent system, or a consistent system of dependent equations. Then solve each system using a graph or a table.

14. Consistent or inconsistent system?

Dependent or independent equations?

Solution: ____________________

5 2 10

55

2

x y

y x

Use the slope-intercept form of each line to determine the number of solutions of the system and then classify each system as a consistent system of independent equations, an inconsistent system, or a consistent system of dependent equations. Then solve each system using a graph or a table.

15. Consistent or inconsistent system?

Dependent or independent equations?

Solution: ____________________

8

4

x y

x y

Use the slope-intercept form of each line to determine the number of solutions of the system and then classify each system as a consistent system of independent equations, an inconsistent system, or a consistent system of dependent equations. Then solve each system using a graph or a table.

16. The tables below display the charges for each service based upon the number of miles driven. Service A has an initial charge of $2.30 and $0.15 for each quarter mile, while Service B has an initial charge of $2.00 and $0.20 for each quarter mile. Service Ax

Milesy

$ Cost

0.50 2.60

1.00 2.90

1.50 3.20

2.00 3.50

2.50 3.80

3.00 4.10

Service Bx

Milesy

$ Cost

0.50 2.40

1.00 2.80

1.50 3.20

2.00 3.60

2.50 4.00

3.00 4.40

(a) Give the solution of the corresponding system of equations.

(b) Interpret the meaning of the x- and y-coordinates of this solution.

Service Ax

Milesy

$ Cost

0.50 2.60

1.00 2.90

1.50 3.20

2.00 3.50

2.50 3.80

3.00 4.10

Service Bx

Milesy

$ Cost

0.50 2.40

1.00 2.80

1.50 3.20

2.00 3.60

2.50 4.00

3.00 4.40

2y

17. At the beginning of the semester a student receives a flier for two different phone plans. The first company offers a rate of 8 cents per minute with no additional fees. The second company offers a rate of 4 cents per minute but has a $20 per month fee in addition to any minutes used. In the past, this student has not used more than 800 minutes in a month. Write an equation to represent the cost y of each plan for one month in which x minutes were used. Company A: 1y

Company B:

17. (a) Use a calculator to graph the two equations. Use the Zoom Fit option to help you determine an appropriate viewing window. Draw a rough sketch of your calculator graph below. See Calculator Perspective 3.3.2 for help.

y

x

17. (b) Use a calculator to find the solution of the system.

(c) Interpret the meaning of the x- and y-coordinates of this solution.