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 1: Check possible solutions of a linear system.

1. Determine whether the ordered pair is a solution of the system of linear equations

2. Determine whether the ordered pair is a solution of the system of linear equations

3. Consider the graph of the system: (a) Determine the point of intersection. (b) Verify that this point satisfies both equations. Objective 2: Solve a system of linear equations by using graphs and tables.

4. Use the table to solve the system of equations. Solution: __________________

5. Solve the system of equations by graphing each line. Solution: __________________

6. Solve the system of linear equations by using a calculator to graph each line and express both coordinates in fraction form. See Technology Perspective for help. Solution: __________________

Objective 3: 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 _____________________. 7. Solve each equation and identify its type from the list above. (a) (b) (c)

8. Sketch the graphs of the lines in each system below. (a)

(b) 8. Sketch the graphs of the lines in each system below.

(c) 8. Sketch the graphs of the lines in each system below.

9.A solution of a system of equations is an __________________, (x, y), that satisfies each equation in the system.

Classify each system as one of the following: (Hint: Examine the graphs shown for problem 8.)TYPO says problem 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. (a) (b) (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.

11. Consistent or inconsistent system? Dependent or independent equations? Solution: ____________________

12. Consistent or inconsistent system? Dependent or independent equations? Solution: ____________________

13. Consistent or inconsistent system? Dependent or independent equations? Solution: ____________________

Consistent or inconsistent system? Dependent or independent equations? Solution: ____________________ 14.

15. 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 A x Miles y $ Cost Service B x Miles y $ Cost

(a) Give the solution of the corresponding system of equations. (b) Interpret the meaning of the x- and y-coordinates of this solution. Service A x Miles y $ Cost Service B x Miles y $ Cost

16. 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: Company B: (a) Determine the restrictions on the variable x.

16. (b) 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 Technology Perspective for help.

(c) Use a calculator to find the solution of the system. (d) Interpret the meaning of the x- and y-coordinates of this solution.