Practice and Problem-Solving Exercises

A Practice

See Problem 1.

Solve each system by graphing. Check your answers.

8. { y = − x 2 + 2 x + 1 y = 2 x + 1 left brace . table with 2 rows and 1 column , row1 column 1 , y equals negative , x squared , plus 2 x plus 1 , row2 column 1 , y equals 2 x plus 1 , end table
9. { y = x 2 − 2 x + 1 y = 2 x + 1 left brace . table with 2 rows and 1 column , row1 column 1 , y equals , x squared , minus 2 x plus 1 , row2 column 1 , y equals 2 x plus 1 , end table
10. { y = x 2 − x + 3 y = − 2 x + 5 left brace . table with 2 rows and 1 column , row1 column 1 , y equals , x squared , minus x plus 3 , row2 column 1 , y equals negative 2 x plus 5 , end table
11. { y = 2 x 2 + 3 x + 1 y = − 2 x + 1 left brace . table with 2 rows and 1 column , row1 column 1 , y equals 2 , x squared , plus 3 x plus 1 , row2 column 1 , y equals negative 2 x plus 1 , end table
12. { y = − x 2 − 3 x + 2 y = x + 6 left brace . table with 2 rows and 1 column , row1 column 1 , y equals negative , x squared , minus 3 x plus 2 , row2 column 1 , y equals x plus 6 , end table
13. { y = − x 2 − 2 x − 2 y = x − 4 left brace . table with 2 rows and 1 column , row1 column 1 , y equals negative , x squared , minus 2 x minus 2 , row2 column 1 , y equals x minus 4 , end table

See Problem 2.

Solve each system by substitution. Check your answers.

14. { y = x 2 + 4 x + 1 y = x + 1 left brace . table with 2 rows and 1 column , row1 column 1 , y equals , x squared , plus 4 x plus 1 , row2 column 1 , y equals x plus 1 , end table
15. { y = − x 2 + 2 x + 10 y = x + 4 left brace . table with 2 rows and 1 column , row1 column 1 , y equals negative , x squared , plus 2 x plus 10 , row2 column 1 , y equals x plus 4 , end table
16. { y = − x 2 + x − 1 y = − x − 1 left brace . table with 2 rows and 1 column , row1 column 1 , y equals negative , x squared , plus x minus 1 , row2 column 1 , y equals negative x minus 1 , end table
17. { y = 2 x 2 − 3 x − 1 y = x − 3 left brace . table with 2 rows and 1 column , row1 column 1 , y equals 2 , x squared , minus 3 x minus 1 , row2 column 1 , y equals x minus 3 , end table
18. { y = x 2 − 3 x − 20 y = − x − 5 left brace . table with 2 rows and 1 column , row1 column 1 , y equals , x squared , minus 3 x minus 20 , row2 column 1 , y equals negative x minus 5 , end table
19. { y = − x 2 − 5 x − 1 y = x + 2 left brace . table with 2 rows and 1 column , row1 column 1 , y equals negative , x squared , minus 5 x minus 1 , row2 column 1 , y equals x plus 2 , end table

See Problem 3.

Solve each system.

20. { y = x 2 + 5 x + 1 y = x 2 + 2 x + 1 left brace . table with 2 rows and 1 column , row1 column 1 , y equals , x squared , plus 5 x plus 1 , row2 column 1 , y equals , x squared , plus 2 x plus 1 , end table
21. { y = x 2 − 2 x − 1 y = − x 2 − 2 x − 1 left brace . table with 2 rows and 1 column , row1 column 1 , y equals , x squared , minus 2 x minus 1 , row2 column 1 , y equals negative , x squared , minus 2 x minus 1 , end table
22. { y = − x 2 − 3 x − 2 y = x 2 + 3 x + 2 left brace . table with 2 rows and 1 column , row1 column 1 , y equals negative , x squared , minus 3 x minus 2 , row2 column 1 , y equals , x squared , plus 3 x plus 2 , end table
23. { y = − x 2 − x − 3 y = 2 x 2 − 2 x − 3 left brace . table with 2 rows and 1 column , row1 column 1 , y equals negative , x squared , minus x minus 3 , row2 column 1 , y equals 2 , x squared , minus 2 x minus 3 , end table
24. { y = − 3 x 2 − x + 2 y = x 2 + 2 x + 1 left brace . table with 2 rows and 1 column , row1 column 1 , y equals negative 3 , x squared , minus x plus 2 , row2 column 1 , y equals , x squared , plus 2 x plus 1 , end table
25. { y = x 2 + 2 x + 1 y = x 2 + 2 x − 1 left brace . table with 2 rows and 1 column , row1 column 1 , y equals , x squared , plus 2 x plus 1 , row2 column 1 , y equals , x squared , plus 2 x minus 1 , end table

See Problem 4.

Solve each system by graphing.

26. { y > x 2 + 2 x y > x 2 − 1 left brace . table with 2 rows and 1 column , row1 column 1 , y greater than , x squared , plus 2 x , row2 column 1 , y greater than , x squared , minus 1 , end table
27. { y > x 2 − 3 x y > 2 x 2 − 3 x left brace . table with 2 rows and 1 column , row1 column 1 , y greater than , x squared , minus 3 x , row2 column 1 , y greater than 2 , x squared , minus 3 x , end table
28. { y < − x 2 − 3 x y > x 2 − 1 left brace . table with 2 rows and 1 column , row1 column 1 , y less than negative , x squared , minus 3 x , row2 column 1 , y greater than , x squared , minus 1 , end table

B Apply

29. Think About a Plan A manufacturer is making cardboard boxes by cutting out four equal squares from the corners of the rectangular piece of cardboard and then folding the remaining part into a box. The length of the cardboard piece is 1 in. longer than its width. The manufacturer can cut out either 3 × 3 3 times 3  in. squares, or 4 × 4 4 times 4  in. squares. Find the dimensions of the cardboard for which the volume of the boxes produced by both methods will be the same.
    • How can you represent the volume of the box using one variable?
    • What system of equations can you write?
    • Which method can you use to solve the system?
30. Open-Ended Can you solve the system of equations shown by graphing? Justify your answer. Can you solve this system using another method? If so, solve the system and explain why you chose that method. { x = y 2 + 2 y + 1 y = x − 4 left brace . table with 2 rows and 1 column , row1 column 1 , x equals , y squared , plus 2 y plus 1 , row2 column 1 , y equals x minus 4 , end table

Solve each system by substitution.

31. { x + y = 3 y = x 2 − 8 x − 9 left brace . table with 2 rows and 1 column , row1 column 1 , x plus y equals 3 , row2 column 1 , y equals , x squared , minus 8 x minus 9 , end table
32. { y − 2 x = x + 5 y + 1 = x 2 + 5 x + 3 left brace . table with 2 rows and 1 column , row1 column 1 , y minus 2 x equals x plus 5 , row2 column 1 , y plus 1 equals , x squared , plus 5 x plus 3 , end table
33. { y − 1 2 x 2 = 1 + 3 x y + 1 2 x 2 = x left brace . table with 2 rows and 1 column , row1 column 1 , y minus , 1 half , x squared , equals 1 plus 3 x , row2 column 1 , y plus , 1 half , x squared , equals x , end table
34. { x + y − 2 = 0 x 2 + y − 8 = 0 left brace . table with 2 rows and 1 column , row1 column 1 , x plus y minus 2 equals 0 , row2 column 1 , x squared , plus y minus 8 equals 0 , end table
35. { x 2 − y = x + 4 x − 1 = y + 3 left brace . table with 2 rows and 1 column , row1 column 1 , x squared , minus y equals x plus 4 , row2 column 1 , x minus 1 equals y plus 3 , end table
36. { 2 y = y − x 2 + 1 y = x 2 − 5 x − 2 left brace . table with 2 rows and 1 column , row1 column 1 , 2 y equals y minus , x squared , plus 1 , row2 column 1 , y equals , x squared , minus 5 x minus 2 , end table

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