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?

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

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

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

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

   • Add rows (1) and (3)
   • Add 2 × row (1) to row (3)
   • Subtract row (3) from 2 × row (1)
   • Subtract row (3) from row (1)
5. 4.

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

   • -3y − 6z = -16
   • 4x − 3y + 10z = -16
   • 13y − 6z = 5
   • 13y − 6z = 12
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
   • -8y − 7z = -46
   • 16y − 7z = 18
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 = 16
   • -11y − 12z = -36
   • 5y + 4z = -23
8. 7.

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

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

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 − 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?

    • Exactly one solution
    • No solution
    • Infinitely many solutions

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?

    • Infinitely many solutions
    • Exactly one solution
    • No solution