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 + y − 2z = 17; (2) 3x − 3y + 3z = -9; (3) -2x + 2y + 3z = -14.

   	x	y	z	total
   (1)	1	1	-2	17
   (2)	3	-3	3	-9
   (3)	-2	2	3	-14

2. 1.

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

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

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

   • y = 5 and z = -4
   • (-1, 5, -4)
   • (-4, 5, -1)
   • (8, -5, -11)
4. 3.

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

   • 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)
5. 4.

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

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

   • 4x − 3y + 10z = -16
   • -3y − 6z = -16
   • 13y − 6z = 12
   • 13y − 6z = 5
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
   • 6x − 8y + 5z = -46
   • 16y − 7z = 18
8. 7.

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

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

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. 8.

    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 − 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.

12. 9.

    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

13. 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.

14. 10.

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

    • Exactly one solution
    • Infinitely many solutions
    • No solution