1. Day 84 – Proving Elimination

2. Vocabulary
A system of equations that has at least one solution is called a CONSISTENT system. A system of equations that has exactly one solution is called an INDEPENDENT system. A system of equations that has no solution is called an INCONSISTENT system. A system of equations that has an infinite number of solutions is called a DEPENDENT system

3. Examples
Use a graphing utility to determine the number of solutions.

4. Solution
Graph the equations. Notice that the lines have the same slope but different y-intercepts. Lines with the same slope but different y-intercepts are parallel and cannot intersect. Therefore, the system has no solution. If you try to solve this system using elimination, the result is a false equation. The false equations tells you that the system is inconsistent

5. Example 2
Use a graphing utility to determine the number of solutions.

6. Solution
Write each equation in slope-intercept form
Graph the equations. The system consists of two equations, one a multiple of the other. Any point on the graph of x + 2y = 4 is also on the graph of 2x + 4y = 8. Therefore, any ordered pair that satisfies one equation will also satisfy the other equation. There are an infinite number of solutions to this linear system.
A system of equations with the same graph is called a dependent system. Because such a system has at least one solution it is also Consistent.

7. if you try to solve this system using elimination, the result is the equation 0 = 0. A true equation, like 0 = 0 or ─3 = ─3, tells you that a linear system is dependent.

8. Number of Units Sold, in hundred thousands
Examples 3
The sales department at Synergy Computers collected the following data about the growth of Synergy and of the firm's chief competitor, Compco. If the companies maintain their current rates of growth, when will Synergy overtake Compco in sales?
Number of Units Sold, in hundred thousands
1991
1992
1993
1994
Synergy
2.3
2.7
3.1
3.5
Compco
2.9
3.3
3.7
4.1

9. Solution
Let x represent the year number (year 1991 = 0). Let y represent the number of units sold in hundred thousands. You can use linear regression to determine these two equations. You can also use other methods you learned in Chapter 6 to write an equation of a line, given two points.

10. The graphs of the equations are shown at the right
The graphs of the equations are shown at the right. The slopes are equal, so the lines are parallel. The parallel lines indicate that the two companies are growing at the same rate. If the growth rates do not change, Synergy will never overtake Compco in sales

11. Writing a System of Linear Equations
Work with a partner. 1. Your cousin is 3 years older than you. Your ages can be represented by two linear equations. Your age Your cousin’s age

12. a. Graph both equations in the same coordinate plane.

13. a. Graph both equations in the same coordinate plane.

14. b. What is the vertical distance between the two graphs
b. What is the vertical distance between the two graphs? What does this distance represent? c. Do the two graphs intersect? If not, what does this mean in terms of your age and your cousin’s age?

15. b. What is the vertical distance between the two graphs
b. What is the vertical distance between the two graphs? What does this distance represent? 3 units, 3 years c. Do the two graphs intersect? If not, what does this mean in terms of your age and your cousin’s age? No, my cousin will always be 3 years older than me

16. Using A Table To Solve A System
Work with a partner. 1. You invest $500 for equipment to make dog backpacks. Each backpack costs you $15 for materials. You sell each backpack for $15. a. Copy and complete the table for your cost C and your revenue R. b. When will your company break even? What is wrong? x 1 2 3 4 5 6 7 8 9 10 C R

17. 1. You invest $500 for equipment to make dog backpacks
1. You invest $500 for equipment to make dog backpacks. Each backpack costs you $15 for materials. You sell each backpack for $15. a. Copy and complete the table for your cost C and your revenue R. b. When will your company break even? What is wrong? Never. Both numbers are growing up (slope) by $15. The revenue will never catch up to the cost. x 1 2 3 4 5 6 7 8 9 10 C
500
515
530
545
560
575
590
605
620
635
650 R 15 30 45 60 75 90
105
120
135
150

18. Using A Graph To Solve A Puzzle
Work with partner. 1. Let x and y be two numbers. Here are two clues about the values of x and y.
Words
Equation
Clue 1:
Y is 4 more than twice the value of x
y = 2x + 4
Clue 2:
The difference of 3y and 6x is 12
3y ─ 6x = 12

19. Using A Graph To Solve A Puzzle
Work with partner. 1. Let x and y be two numbers. Here are two clues about the values of x and y.
Words
Equation
Clue 1:
Y is 4 more than twice the value of x
y = 2x + 4
Clue 2:
The difference of 3y and 6x is 12
3y ─ 6x = 12
y = 2x + 4

20. a. Graph both equations in the same coordinate plane.
b. Do the two lines intersect? Explain.
c. What is the solution of the puzzle?

21. a. Graph both equations in the same coordinate plane.
b. Do the two lines intersect? Explain.
 yes at every point
c. What is the solution of the puzzle?
All the solutions on the line
y = 2x +4

22. d. Use the equation y = 2x + 4 to complete the table. e
d. Use the equation y = 2x + 4 to complete the table. e. Does each solution in the table satisfy both clues? f. What can you conclude? How many solutions does the puzzle have? How can you describe them? X 1 2 3 4 5 6 7 8 9 10 Y

23. d. Use the equation y = 2x + 4 to complete the table. e
d. Use the equation y = 2x + 4 to complete the table. e. Does each solution in the table satisfy both clues? yes f. What can you conclude? How many solutions does the puzzle have? How can you describe them? Infinite solutions on the line y = 2x + 4 X 1 2 3 4 5 6 7 8 9 10 Y 12 14 16 18 20 22 24