Worksheet: Solving Systems of Quadratic Equations

In this worksheet, we will practice finding values that satisfy two quadratic equations in two variables simultaneously.

Q1:

Find all the real solutions to the following system of equations 𝑦=βˆ’2π‘₯+12π‘₯βˆ’13,𝑦=βˆ’π‘₯3+2π‘₯+2.

  • Aπ‘₯=βˆ’3 and 𝑦=βˆ’67
  • BThere are no solutions.
  • Cπ‘₯=9 and 𝑦=βˆ’67
  • Dπ‘₯=3 and 𝑦=5
  • Eπ‘₯=6 and 𝑦=βˆ’13

Q2:

Find all the real solutions to the system of equations 3π‘¦βˆ’π‘₯+3π‘₯+5=0,2π‘¦βˆ’π‘₯+π‘₯+1=0.

  • Aπ‘₯=2 or π‘₯=βˆ’2, 𝑦=1 or 𝑦=5
  • Bπ‘₯=4 or π‘₯=βˆ’4, 𝑦=3 or 𝑦=11
  • CThere are no solutions.
  • Dπ‘₯=4 and 𝑦=11
  • Eπ‘₯=16 and 𝑦=βˆ’7

Q3:

Find all the real solutions to the system of equations: 4𝑦+π‘₯βˆ’4π‘₯βˆ’40=0,2π‘¦βˆ’72π‘₯βˆ’6π‘₯+4=0.

  • Aπ‘₯=βˆ’2.5 and 𝑦=5.9375, π‘₯=2.2 and 𝑦=11.29
  • Bπ‘₯=βˆ’3 and 𝑦=4.75, π‘₯=2 and 𝑦=11
  • Cπ‘₯=βˆ’2.5 and 𝑦=βˆ’5.9375, π‘₯=2.2 and 𝑦=βˆ’10.99
  • Dπ‘₯=βˆ’2 and 𝑦=7, π‘₯=3 and 𝑦=10.75
  • EThere are no solutions.

Q4:

Find all the real solutions to the system of equations 5π‘¦βˆ’π‘₯+9π‘₯βˆ’24=0,3𝑦+π‘₯βˆ’9π‘₯+18=0.

  • Aπ‘₯=4.5 and 𝑦=βˆ’3
  • Bπ‘₯=4.5 and 𝑦=0.75
  • CThere are no solutions.
  • Dπ‘₯=8.7 and 𝑦=βˆ’5.25
  • Eπ‘₯=12.6 and 𝑦=βˆ’21

Q5:

Find all the real solutions to the system of equations 𝑦=π‘₯βˆ’3π‘₯βˆ’4,𝑦=2π‘₯βˆ’4π‘₯βˆ’6.

  • Aπ‘₯=1 and 𝑦=0, π‘₯=βˆ’2 and 𝑦=6
  • BThere are no solutions.
  • Cπ‘₯=1 and 𝑦=0, π‘₯=2 and 𝑦=βˆ’6
  • Dπ‘₯=βˆ’1 and 𝑦=0, π‘₯=2 and 𝑦=βˆ’6
  • Eπ‘₯=βˆ’1 and 𝑦=0, π‘₯=103 and 𝑦=βˆ’269

Q6:

Find all the real solutions to the system of equations 𝑦=π‘₯βˆ’7,𝑦=(π‘₯+1)(π‘₯βˆ’5).

  • Aπ‘₯=βˆ’3 and 𝑦=2
  • Bπ‘₯=βˆ’0.5 and 𝑦=βˆ’6.75
  • Cπ‘₯=0.5 and 𝑦=βˆ’6.75
  • Dπ‘₯=3 and 𝑦=2
  • Eπ‘₯=βˆ’2 and 𝑦=βˆ’3

Q7:

Find all the real solutions to the system of equations 𝑦=3π‘₯+5π‘₯+2,𝑦=βˆ’2π‘₯βˆ’2π‘₯+2.

  • AThere are no solutions.
  • Bπ‘₯=0 and 𝑦=2, π‘₯=βˆ’7 and 𝑦=114
  • Cπ‘₯=0 and 𝑦=2, π‘₯=βˆ’0.6 and 𝑦=0.08
  • Dπ‘₯=1 and 𝑦=βˆ’2, π‘₯=βˆ’1.4 and 𝑦=0.88
  • Eπ‘₯=0 and 𝑦=2, π‘₯=βˆ’1.4 and 𝑦=0.88

Q8:

Find all the real solutions to the system of equations 𝑦=βˆ’2π‘₯+4π‘₯+6,𝑦=4π‘₯+22π‘₯+24.

  • Aπ‘₯=βˆ’1.15 and 𝑦=βˆ’1.245, π‘₯=βˆ’7.85 and 𝑦=97.79
  • Bπ‘₯=βˆ’1.5 and 𝑦=βˆ’4.5, π‘₯=βˆ’4 and 𝑦=βˆ’42
  • Cπ‘₯=βˆ’1.3 and 𝑦=βˆ’2.58, π‘₯=βˆ’11.7 and 𝑦=βˆ’64.2
  • DThere are no solutions.
  • Eπ‘₯=βˆ’1 and 𝑦=6, π‘₯=3 and 𝑦=126

Q9:

Use the graph to solve the system of equations

𝑦=3π‘₯+36π‘₯+96,𝑦=βˆ’5π‘₯+10π‘₯+40.

  • Aπ‘₯=0 and 𝑦=40,π‘₯=βˆ’7.5 and 𝑦=βˆ’10
  • Bπ‘₯=0 and 𝑦=40
  • CThere are no solutions.
  • Dπ‘₯=βˆ’4 and 𝑦=0,π‘₯=βˆ’8 and 𝑦=0
  • Eπ‘₯=βˆ’7.5 and 𝑦=βˆ’10

Q10:

Use the graph to solve the system of equations 𝑦=π‘₯βˆ’10π‘₯+32,𝑦=βˆ’2π‘₯+20π‘₯βˆ’43.

  • Aπ‘₯=7 and 𝑦=5
  • Bπ‘₯=6 and 𝑦=10
  • Cπ‘₯=4 and 𝑦=5
  • Dπ‘₯=10 and 𝑦=6
  • Eπ‘₯=5 and 𝑦=7

Q11:

Use the graph to solve the system of equations

𝑦=3π‘₯+2π‘₯βˆ’8,𝑦=π‘₯+4π‘₯+4.

  • AThere are no solutions.
  • Bπ‘₯=2 and 𝑦=0, π‘₯=3 and 𝑦=25
  • Cπ‘₯=βˆ’2 and 𝑦=0, π‘₯=3 and 𝑦=25
  • Dπ‘₯=0 and 𝑦=2, π‘₯=25 and 𝑦=3
  • Eπ‘₯=0 and 𝑦=βˆ’2, π‘₯=25 and 𝑦=3

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