systems of three equations: solve, and count the solutions

Grade 11 Math Worksheet

Circle the best answer.

1. Read this first

   Use the table

   Equations: (1) -3x + 2y + z = 12; (2) 3x − 4y − z = -4; (3) 6x + 5y + 2z = -40.

   	x	y	z	total
   (1)	-3	2	1	12
   (2)	3	-4	-1	-4
   (3)	6	5	2	-40

2. 1.

   Luz cancelled z with rows (1) and (2). Row (3) has a z number twice the size of row (1)'s. Which step on rows (1) and (3) cancels z?

   • Add rows (1) and (3)
   • Subtract row (3) from row (1)
   • Add 2 × row (1) to row (3)
   • Subtract row (3) from 2 × row (1)
3. 2.

   Noah cancelled z from rows (1) and (2) and now works on rows (1) and (3). Which step cancels z?

   • Subtract row (3) from row (1)
   • Subtract row (3) from 2 × row (1)
   • Add rows (1) and (3)
   • Add 2 × row (1) to row (3)
4. 3.

   Pia multiplies row (1) by 2 and subtracts it from row (2) to cancel x. Which new row is left?

   • 13y − 6z = 5
   • 4x − 3y + 10z = -16
   • -3y − 6z = -16
   • 13y − 6z = 12
5. 4.

   To cancel x, Ren subtracts 3 × row (1) from row (2) of the number table. Which new row is left?

   • -8y − 7z = -46
   • 16y − 7z = 50
   • 6x − 8y + 5z = -46
   • 16y − 7z = 18
6. 5.

   In Ari's table, row (2) minus 2 × row (1) cancels x. Which new row is left?

   • 9y + 3z = 60
   • 4x − 3y − 13z = -72
   • 9y + 3z = 27
   • -3y − 13z = -72

7. Use the table

   Equations: (1) 2x + 2y + z = -8; (2) -5x − 4y − z = 6; (3) 4x + 2y − z = 11.

   	x	y	z	total
   (1)	2	2	1	-8
   (2)	-5	-4	-1	6
   (3)	4	2	-1	11

   Row (4) = row (1) + row (2): -3x − 2y = -2. Row (5) = row (1) + row (3): 6x + 4y = 3.

8. 6.

   Bo adds row (1) to rows (2) and (3) to cancel z, giving rows (4) and (5). How many solutions does the system have?

   • No solution
   • Infinitely many solutions
   • Exactly one solution

9. Use the table

   Equations: (1) 2x + 3y − z = 3; (2) -6x − 4y + z = -10; (3) 2x − 2y + z = 4.

   	x	y	z	total
   (1)	2	3	-1	3
   (2)	-6	-4	1	-10
   (3)	2	-2	1	4

   Row (4) = row (1) + row (2): -4x − y = -7. Row (5) = row (1) + row (3): 4x + y = 7.

10. 7.

    Esme cancels z, making rows (4) and (5) from the table. How many solutions does the system have?

    • Exactly one solution
    • Infinitely many solutions
    • No solution

11. Use the table

    Equations: (1) 3x − 2y + z = -3; (2) -7x + 3y − z = 8; (3) -2x + 6y − z = -11.

    	x	y	z	total
    (1)	3	-2	1	-3
    (2)	-7	3	-1	8
    (3)	-2	6	-1	-11

    Row (4) = row (1) + row (2): -4x + y = 5. Row (5) = row (1) + row (3): x + 4y = -14.

12. 8.

    Adding row (1) to rows (2) and (3) gives Jun rows (4) and (5) without z. How many solutions does the system have?

    • Exactly one solution
    • No solution
    • Infinitely many solutions

13. Use the table

    Equations: (1) -x − 2y + 3z = -5; (2) -x − 2y − 3z = 1; (3) -3x − 6y − 3z = -4.

    	x	y	z	total
    (1)	-1	-2	3	-5
    (2)	-1	-2	-3	1
    (3)	-3	-6	-3	-4

    Row (4) = row (1) + row (2): -2x − 4y = -4. Row (5) = row (1) + row (3): -4x − 8y = -9.

14. 9.

    Kira adds row (1) to rows (2) and (3) to cancel z, giving rows (4) and (5). How many solutions does the system have?

    • Exactly one solution
    • Infinitely many solutions
    • No solution

15. Use the table

    Equations: (1) x − y − 3z = -2; (2) 2x + 2y + 3z = 4; (3) 8x + 4y + 3z = 8.

    	x	y	z	total
    (1)	1	-1	-3	-2
    (2)	2	2	3	4
    (3)	8	4	3	8

    Row (4) = row (1) + row (2): 3x + y = 2. Row (5) = row (1) + row (3): 9x + 3y = 6.

16. 10.

    Luz cancels z, making rows (4) and (5) from the table. How many solutions does the system have?

    • No solution
    • Exactly one solution
    • Infinitely many solutions