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Grade 9 · Math · Algebra 1

9th grade Systems: one, none or infinitely many solutions, and the quickest method worksheet

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Systems: one, none or infinitely many solutions, and the quickest method

Grade 9 Math Worksheet - Challenge Me

NameDate

Circle the best answer, or write your answer on the line.

  1. Ren writes 3x + 4y = 10 and 6x + ky = 15. For which k does the system have no solution?

  2. Tala writes 2x + 5y = 7 and 6x + 15y = c. For which c does the system have infinitely many solutions?

  3. Zee writes 4x + 6y = 10 and 6x + ky = 15. For which k does the system have infinitely many solutions?

  4. Ari compares y = 2(x + 4) and y - 2x = 8. How many solutions does the system have?

    • Infinitely many solutions (dependent)
    • One solution (independent), because the lines cross once
    • No solution (inconsistent)

  5. Bo compares y = 3(x - 2) and y = 3x - 6. How many solutions does the system have?

    • No solution (inconsistent), since the two rules look different
    • One solution (independent)
    • Infinitely many solutions (dependent)

  6. Esme checks y = 2(x + 3) and y = 2x + 3. How many solutions does the system have?

    • No solution (inconsistent)
    • One solution (independent)
    • Infinitely many solutions (dependent)

  7. Jun checks y = 2(x + 3) - 1 and y = 2x + 2. How many solutions does the system have?

    • No solution (inconsistent)
    • One solution (independent)
    • Infinitely many solutions (dependent)

  8. Read this first

    Old system

    2x + y = 11 and x - y = 1. New system: 5x - 2y = 14 and x - y = 1.

  9. Luz replaces the first equation with "first + 3 × second". The old solution is (4, 3). Which point solves the new system?

    • (4, 3)
    • (2, -2)
    • (3, 4)
    • (12, 9)

  10. Old system

    3x + 2y = 16 and 2x - y = 6. New system: 7x = 28 and 2x - y = 6.

  11. Noah replaces the first equation with "first + 2 × second". The old solution is (4, 2). Which point solves the new system?

    • (4, 2)
    • (4, 0)
    • (8, 4)
    • (2, 4)

  12. Old system

    x + 3y = 7 and 2x - y = 7. New system: -3x + 5y = -7 and 2x - y = 7.

  13. Pia replaces the first equation with "first - 2 × second". The old solution is (4, 1). Which point solves the new system?

    • (-1, -2)
    • (4, 1)
    • (8, 2)
    • (1, 4)
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9th grade Systems: one, none or infinitely many solutions, and the quickest method worksheet – printable
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About this worksheet

Systems: how many solutions, and the quickest method. Every version has 10 questions, and the answer key prints on the second page.

What your child will learn

  • I can tell, without graphing, whether a system has one solution, no solution or infinitely many, by comparing the step and the start of each rule.
  • I can rewrite an equation to y = mx + b, or compare ax + by = c pairs, and choose the first step that makes a letter disappear.
  • I can find the number that makes a system have no solution, and I know a solution stays the same when one equation is replaced by the sum of two.

Standard: A-REI.C.5, A-REI.C.6

Words to know

system: Two equations about the same x and y.

independent: Different steps: the lines cross once, so one solution.

inconsistent: Same step, different starts: the lines never meet, so no solution.

dependent: Same step and same start: one line twice, so infinitely many solutions.

step / start: The step is how much y changes for 1 more x; the start is y when x is 0.

eliminate: Add or subtract equations so one letter disappears.

Common mistakes

Your child expected two lines to cross once, even though the steps match.

Your child saw two equally steep lines and called them the same line.

Your child said two matching rules never meet, but they are the same line.

What to say

Ask: if two lines climb at the same rate, can the gap between them ever shrink?

Ask: do the two rules start at the same height?

Ask: do the rules give the same y for x = 0 and for x = 1?

Questions parents ask

What is the most common mistake with systems: one, none or infinitely many solutions, and the quickest method?

Your child expected two lines to cross once, even though the steps match.

How can I help my child with systems: one, none or infinitely many solutions, and the quickest method?

Ask: if two lines climb at the same rate, can the gap between them ever shrink?

Are these 9th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

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