Systems of Equations practice questions

Systems of Equations questions

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1. 1Systems of equationsStandard

   If x + y = 10 and x - y = 4, what is the value of x?If x + y = 10 and x - y = 4, what is the value of x?

   • A55
   • B33
   • C1414
   • D77
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   Answer: 77

   The y terms are already opposites, so adding the equations eliminates y: 2x = 14, and x = 7. Check for that before substituting; it is faster whenever a variable appears with matching coefficients.The y terms are already opposites, so adding the equations eliminates y: 2x = 14, and x = 7. Check for that before substituting; it is faster whenever a variable appears with matching coefficients.

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2. 2Systems of equationsStandard

   If y = 3x - 2 and 2x + y = 13, what is the value of x?If y = 3x - 2 and 2x + y = 13, what is the value of x?

   • A55
   • B44
   • C33
   • D22
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   Answer: 33

   Substitute 3x - 2 for y: 2x + 3x - 2 = 13, so 5x = 15 and x = 3.Substitute 3x - 2 for y: 2x + 3x - 2 = 13, so 5x = 15 and x = 3.

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3. 3Systems of equationsStandard

   If 3x + 2y = 16 and 3x - 2y = 4, what is the value of y?If 3x + 2y = 16 and 3x - 2y = 4, what is the value of y?

   • A66
   • B33
   • C44
   • D22
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   Answer: 33

   Subtract the second equation from the first: (3x + 2y) - (3x - 2y) = 16 - 4 gives 4y = 12, so y = 3. Elimination is cheaper here than substitution because the x terms already match.Subtract the second equation from the first: (3x + 2y) - (3x - 2y) = 16 - 4 gives 4y = 12, so y = 3. Elimination is cheaper here than substitution because the x terms already match.

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4. 4Systems of equationsStandard

   If 2x + 5y = 26 and x = y + 1, what is the value of y?If 2x + 5y = 26 and x = y + 1, what is the value of y?

   • A26726/7
   • B33
   • C24724/7
   • D44
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   Answer: 24724/7

   Substitute x = y + 1: 2(y + 1) + 5y = 26, so 7y + 2 = 26 and 7y = 24, giving y = 247.Substitute x = y + 1: 2(y + 1) + 5y = 26, so 7y + 2 = 26 and 7y = 24, giving y = 24/7.

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5. 5Systems of equationsStandard

   If 4x - y = 11 and x + y = 4, what is the value of x?If 4x - y = 11 and x + y = 4, what is the value of x?

   • A55
   • B33
   • C22
   • D11
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   Answer: 33

   Adding the equations eliminates y: 5x = 15, so x = 3. Elimination is cheaper here than substitution because the y terms already cancel.Adding the equations eliminates y: 5x = 15, so x = 3. Elimination is cheaper here than substitution because the y terms already cancel.

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6. 6Systems of equationsStandard

   If 4x + 3y = 27 and 2x - y = 1, what is the value of x + y?If 4x + 3y = 27 and 2x - y = 1, what is the value of x + y?

   • A88
   • B33
   • C55
   • D22
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   Answer: 88

   The second equation gives y = 2x - 1. Substituting: 4x + 3(2x - 1) = 27, so 10x = 30 and x = 3, making y = 5. Then x + y = 8. The wrong answers are x and y themselves, which is what you get by stopping one step early.The second equation gives y = 2x - 1. Substituting: 4x + 3(2x - 1) = 27, so 10x = 30 and x = 3, making y = 5. Then x + y = 8. The wrong answers are x and y themselves, which is what you get by stopping one step early.

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7. 7Systems of equationsStandard

   If 3x + 2y = 19 and x - y = 3, what is the value of y?If 3x + 2y = 19 and x - y = 3, what is the value of y?

   • A55
   • B33
   • C1919
   • D22
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   Answer: 22

   The second equation gives x = y + 3. Substituting: 3(y + 3) + 2y = 19, so 5y + 9 = 19 and y = 2. The other choices are x, the constant from the second equation, and the total from the first.The second equation gives x = y + 3. Substituting: 3(y + 3) + 2y = 19, so 5y + 9 = 19 and y = 2. The other choices are x, the constant from the second equation, and the total from the first.

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8. 8Systems of equationsStandard

   If 2x + y = 13 and y = x + 1, enter the value of x.If 2x + y = 13 and y = x + 1, enter the value of x.

   Student-produced response: write your own answer rather than choosing one.

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   Answer: 44

   Substituting the second equation into the first gives 2x + x + 1 = 13, so 3x = 12 and x = 4 (with y = 5). Check: 8 + 5 = 13.Substituting the second equation into the first gives 2x + x + 1 = 13, so 3x = 12 and x = 4 (with y = 5). Check: 8 + 5 = 13.

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9. 9Systems of equationsStandard

   If 4x + y = 17 and y = 2x - 1, what is the value of x?If 4x + y = 17 and y = 2x - 1, what is the value of x?

   • A44
   • B22
   • C33
   • D66
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   Answer: 33

   Substituting the second into the first gives 4x + 2x - 1 = 17, so 6x = 18 and x = 3 (with y = 5). Check: 12 + 5 = 17.Substituting the second into the first gives 4x + 2x - 1 = 17, so 6x = 18 and x = 3 (with y = 5). Check: 12 + 5 = 17.

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10. 10Systems of equationsStandard

    If 2x + 3y = 21 and y = x + 2, what is the value of y?If 2x + 3y = 21 and y = x + 2, what is the value of y?

    • A55
    • B33
    • C77
    • D99
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    Answer: 55

    Substituting gives 2x + 3(x + 2) = 21, so 5x + 6 = 21 and x = 3, making y = 5. Check: 6 + 15 = 21. The choice 3 is x rather than y.Substituting gives 2x + 3(x + 2) = 21, so 5x + 6 = 21 and x = 3, making y = 5. Check: 6 + 15 = 21. The choice 3 is x rather than y.

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11. 11Systems of equationsChallenge

    For which value of k does the system 2x + ky = 8 and 4x + 6y = 5 have no solution?For which value of k does the system 2x + ky = 8 and 4x + 6y = 5 have no solution?

    • A22
    • B1212
    • C33
    • D66
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    Answer: 33

    A system has no solution when the two lines are parallel but not identical. Multiplying the first equation by 2 gives 4x + 2ky = 16, which matches 4x + 6y in the x and y terms when 2k = 6, so k = 3. The constants 16 and 5 differ, so the lines never meet.A system has no solution when the two lines are parallel but not identical. Multiplying the first equation by 2 gives 4x + 2ky = 16, which matches 4x + 6y in the x and y terms when 2k = 6, so k = 3. The constants 16 and 5 differ, so the lines never meet.

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12. 12Systems of equationsChallenge

    The system 3x + ky = 9 and 6x + 10y = 18 has infinitely many solutions. What is k?The system 3x + ky = 9 and 6x + 10y = 18 has infinitely many solutions. What is k?

    • A1010
    • B55
    • C33
    • D2020
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    Answer: 55

    Infinitely many solutions means the equations describe the same line. Doubling the first gives 6x + 2ky = 18, which matches the second when 2k = 10, so k = 5. The constants already agree, which is what makes them identical rather than parallel.Infinitely many solutions means the equations describe the same line. Doubling the first gives 6x + 2ky = 18, which matches the second when 2k = 10, so k = 5. The constants already agree, which is what makes them identical rather than parallel.

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13. 13Systems of equationsChallenge

    For which value of c does the system 3x - y = 4 and 6x - 2y = c have infinitely many solutions?For which value of c does the system 3x - y = 4 and 6x - 2y = c have infinitely many solutions?

    • A44
    • B88
    • C22
    • D00
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    Answer: 88

    Infinitely many solutions means the two equations describe the same line. The second is the first multiplied by 2 on the left, so the right must be doubled as well: c = 8. Any other value gives two parallel lines and no solution at all.Infinitely many solutions means the two equations describe the same line. The second is the first multiplied by 2 on the left, so the right must be doubled as well: c = 8. Any other value gives two parallel lines and no solution at all.

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