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) x − 3y + z = -12; (2) -3x − y + 2z = -4; (3) -x + 2y − 3z = 13.

   	x	y	z	total
   (1)	1	-3	1	-12
   (2)	-3	-1	2	-4
   (3)	-1	2	-3	13

2. 1.

   Bo writes a system of three equations as the number table shown. Which ordered triple (x, y, z) solves it?

   • y = 3 and z = -2
   • (-2, 3, -1)
   • (-1, 3, -2)
   • (-6, -2, -12)
3. 2.

   Esme's math club card gives the system in the table. Which ordered triple (x, y, z) solves all three equations?

   • (4, 5, 5)
   • (12, 3, 3)
   • (5, 5, 4)
   • y = 5 and z = 4
4. 3.

   Kira is checking a homework system, shown as a table. Which ordered triple (x, y, z) is its solution?

   • (8, -5, -11)
   • (-4, 5, -1)
   • (-1, 5, -4)
   • y = 5 and z = -4
5. 4.

   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?

   • Subtract row (3) from row (1)
   • Add 2 × row (1) to row (3)
   • Add rows (1) and (3)
   • Subtract row (3) from 2 × row (1)
6. 5.

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

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

7. 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.

8. 6.

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

   • Infinitely many solutions
   • No solution
   • Exactly one solution

9. 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.

10. 7.

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

    • Infinitely many solutions
    • No solution
    • Exactly one solution

11. 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.

12. 8.

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

    • Infinitely many solutions
    • No solution
    • Exactly one solution

13. 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.

14. 9.

    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

15. Use the table

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

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

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

16. 10.

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

    • Infinitely many solutions
    • No solution
    • Exactly one solution