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?

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

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

   • (-4, 4, 5)
   • (2, -5, -10)
   • (5, 4, -4)
   • y = 4 and z = -4
4. 3.

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

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

   Milo needs a second row without z, from rows (1) and (3) of the table. Which step does that?

   • 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.

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

   • Subtract row (3) from 2 × row (1)
   • Add rows (1) and (3)
   • Add 2 × row (1) to row (3)
   • Subtract row (3) from row (1)
7. 6.

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

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

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

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. 8.

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

    • Exactly one solution
    • No solution
    • Infinitely many solutions

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. 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?

    • No solution
    • Exactly one solution
    • Infinitely many solutions

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. 10.

    Luz 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