Activities

1. Determine if the following points are solutions to the first, second, both, or neither of linear equations below.
   
   2x + 3y = 6                  3x – y = 9
   
   a. (1, -6)                       b. (2, 2)                        c. (3, 0)                        d. (-3, 4)

2. Look at the graph below and determine which x and y value is on both of the lines.

3. Two schools are going on a field trip. High school A is traveling in 4 busses and 3 vans. High school B is traveling in 2 busses and 5 vans. 180 students are going on the trip from high school A and 165 students are attending from school B. Write a system of equations that could be used to solve for b, the number of students on each bus, and v the number of students on each van.

4. Given the following three sets of linear equations, describe the technique (graphing, substitution, or elimination) that you would be best to solve each system.

1. y = x – 7, y = -2x + 3

2. 2x + 3y = 11, x = 5 – y

3. 3x + 7y = 30, x – 2y = -3

4. 7y – 14x = 21, y = 1

5. x + y = 10, x – y = -2

5. When solving the following systems using elimination, what constant would you multiply each equation by in order to cancel out one of the variables?

a. ___(x – 5y = 8)                    b. ___(7x – 2y = -2)                c. ___(3x – 5y = 87)
    ___(3x – 5y = 11)                    ___(5x + 3y = 16)                   ___(-5x – y = 110)

d. ___(11x + 6y = 8)               e. ___(9x – 8y = 8)                  f. ___
    ___(-9x – 8y = 11)                  ___(4x – 5y = 11)                   ___

6. Find the solution to the following system of equations by graphing.

and y = 0.5x

7. Solve the system below using substitution.

y = 3x – 5 and x = 7 – y

8. Determine the pair of x and y values that are solutions to both of these equations using elimination.

8x + 14y = 4 and -6x – 7y = -10 

9. Use any method to solve the following system of equations.

0 = 14 + x + 7y and -4x – 14y = 28 

10. Solve the system below by graphing.
    7x – 3.5y = 21 and 2(x – y) = 2x – 4

11. Solve the following system by elimination.
    3x – 3y = 18 and y – x = -2      
                                                                                                                                                                  
12. Two grades at a high school are competing against one another to see who can sell the most tickets. The 9^th graders sell 57 adult tickets and 12 student tickets for a total of $532.50. Meanwhile, the 10^th graders sell 68 adult tickets and 17 student tickets for a total of $646.00. How much does each adult and each student ticket cost?

13. A 40-question test is worth a total of 85 points. The test is made up of true or false questions, worth one point each, and multiple-choice questions, worth 2.5 points each. Determine how many of the questions are true and false, and how many are multiple choice?

14. The sum of two numbers is 97. Their difference is 45. What are the two numbers?
    
    
15. A certain pet store sells only dogs and birds. They have a total of 71 animals in the store currently. A story employee walked around and counted a total of 246 legs. How many of the animals are dogs?

16. Write the equations of two lines such that the system of linear equations has one solution at the coordinate (-3, -4).

17. One pear is 110 calories more than one banana. Two pears are 60 less calories than 6 bananas. How many calories are in one banana and how many calories are in one pear?

18. Show and explain how you would solve the following system of equations:

-5x + 3y = 46                          2y – 39 = 5x

                        SHOW:                                                           EXPLAIN:

19. Gary bought x slices of pizza and y sodas for his friends during lunch

1. Gary purchased a total of 15 items. Write an equation to represent this situation.
2. Pizza costs $3.00 a slice and soda costs $2.00 per bottle. Gary spent a total of $40 while he was out buying lunch for his friends. Write an equation to represent this situation.
3. How many slices of pizza did Gary buy for his friends? Show AND explain your work.

20. Solve the following system of equations using all three methods, graphing, substitution, and elimination.

x – y  = -3

3x – 2y = -4

21. Sasha and her sister Carmen are comparing their bank accounts. Two years ago, Sasha had $500.00 in savings and has been saving an extra $25.00 each month. Looking back on the same timeframe, Carmen had $1025.00 saved and has been spending $50.00 more than she earns each month since then. How many months ago did Sasha and Carmen have the same amount of money? Use graphing to determine your solution.

22. Create a system of equations that has a solution in quadrant 3.

23. Given the two equations, 2x – 3y = 7 and 6x + by = c, what are values of b and c that would cause this system to have:

1. No solutions

2. Infinite solutions

24. A boat was being paddled downstream, with the current. It took 2 hours to travel 12 miles. On the return trip, now traveling against the current, the same distance took 8 hours to travel. What is the speed of the boat in still water and the speed of the current.
    
    
25. Solution A is 50% acid and solution B is 80% acid. How much of each should be used to make 100cc. of a solution that is 68% acid?

Answer Key/Rubric

1. a. Second        b. Neither        c. Both                        d. First

2. (-2, 4)

3. 4b + 3v = 180

2b + 5v = 165

4. Various answers including:
   1. graphing
   2. substitution
   3. elimination
   4. substitution
   5. elimination

5. Various answers including:

a. 1(x – 5y = 8)                        b. 3(7x – 2y = -2)                    c. 1(3x – 5y = 87)
    -1(3x – 5y = 11)                       2(5x + 3y = 16)                       -5(-5x – y = 110)

d. 4(11x + 6y = 8)                   e. 5(9x – 8y = 8)                      f. 2(
    3(-9x – 8y = 11)                      -8(4x – 5y = 11)                      3(

6. (6, 3)

7. (3, 4)

8. (4, -2)

9. (0, -2)

10. (4, 2)

11. No Solution     
                                                                                                                                                                                     
12. Adult ticket $8.50 and student ticket $4.00

13. 30 Multiple choice questions, 10 True or False questions

14. 26 and 71

15. 52 dogs (and 19 birds)

16. Various answers including: y = -4 and y = x – 1

17. One banana is 70 calories, One pear is 180 calories
    
    
18.             SHOW:                                                           EXPLAIN:

-5x + 3y = 46                                                  1. Rewrite the 2^nd equation into standard form
5x  - 2y = -39

y = 7                                                                2. Add the equations together, the x’s cancel

2(7) – 39 = 5x                                                 3. Substitute the value for y into the 2^nd equation

-25 = 5x                                                           4. Solve for x.
x = -5

(-5, 7)                                                              5. Write the solution as a coordinate point

19. Gary bought x slices of pizza and y sodas for his friends during lunch

1. x + y = 15
2. 3x + 2y = 40
3. (10, 5); 10 slices of pizza

20. (2, 5)

21. Various scales on the graph, 7 months

22. Various solutions including: y = -5 and y = 2x – 1

23. 
    
    1. No solutions b = -9, c = any number except 21

2. Infinite solutions b = -9, c = 21

24. The current is moving at 2.25 miles per hour and the boat would move at 3.75 mph in still water

25. 40cc of solution A and 60cc of solution B