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 = 24; (2) x + 2y + 3z = 27; (3) 3x + y − 2z = 12.

   	x	y	z	total
   (1)	1	3	1	24
   (2)	1	2	3	27
   (3)	3	1	-2	12

2. 1.

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

   • (5, 5, 4)
   • y = 5 and z = 4
   • (12, 3, 3)
   • (4, 5, 5)
3. 2.

   Jun copies a system into a number table. Which ordered triple (x, y, z) makes all three rows true?

   • (2, -5, -10)
   • y = 4 and z = -4
   • (-4, 4, 5)
   • (5, 4, -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)
   • (-1, 5, -4)
   • y = 5 and z = -4
   • (-4, 5, -1)
5. 4.

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

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

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

   Tala takes -2 copies of row (1) away from row (2), so x cancels. Which new row is left?

   • -4x − 11y − 12z = -36
   • 5y + 4z = -23
   • 5y + 4z = 16
   • -11y − 12z = -36
8. 7.

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

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

9. 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.

10. 8.

    Bo 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

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. 9.

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

    • No solution
    • Exactly one 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. 10.

    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
    • No solution
    • Infinitely many solutions