Worksheet: Linear-Quadratic Systems of Equations

In this worksheet, we will practice solving linear-quadratic systems of equations.

Q1:

Find all of the solutions to the simultaneous equations 𝑦+π‘₯=7 and 2π‘₯+π‘₯+3𝑦=21.

  • Aπ‘₯=1, 𝑦=5; π‘₯=0, 𝑦=7
  • Bπ‘₯=1, 𝑦=5; π‘₯=0, 𝑦=6
  • Cπ‘₯=1, 𝑦=6; π‘₯=0, 𝑦=7
  • Dπ‘₯=βˆ’2, 𝑦=6; π‘₯=1, 𝑦=7

Q2:

Find the set of points of intersection of the graphs of π‘₯+5𝑦=0 and 𝑦=βˆ’π‘₯.

  • A{(0,0),(βˆ’5,βˆ’25)}
  • B{(0,0),(5,βˆ’25)}
  • C{(0,0),(βˆ’25,5)}
  • D{(0,0),(βˆ’25,βˆ’5)}

Q3:

Given that π‘₯+π‘₯𝑦=18, and π‘₯+𝑦=6, find the value of π‘₯.

Q4:

Find the set of points of intersection of the graphs of π‘₯+𝑦=8 and π‘₯+𝑦=50.

  • A{(7,1),(1,7)}
  • B{(15,7),(9,1)}
  • C{(7,15),(1,9)}
  • D{(7,7),(1,1)}

Q5:

Find all of the solutions to the simultaneous equations 𝑦+2π‘₯=3 and π‘₯+π‘₯𝑦+𝑦=3.

  • Aπ‘₯=2𝑦=βˆ’5π‘₯=1𝑦=1,;,
  • Bπ‘₯=2𝑦=βˆ’1π‘₯=1𝑦=1,;,
  • Cπ‘₯=βˆ’4𝑦=βˆ’1π‘₯=1𝑦=1,;,
  • Dπ‘₯=2𝑦=βˆ’5π‘₯=1𝑦=βˆ’1,;,

Q6:

Find all of the solutions to the simultaneous equations π‘₯βˆ’π‘¦=26 and π‘₯𝑦=120.

  • Aπ‘₯=30𝑦=4π‘₯=βˆ’4𝑦=βˆ’30,;,
  • Bπ‘₯=4𝑦=60π‘₯=βˆ’30𝑦=3,;,
  • Cπ‘₯=βˆ’4𝑦=30π‘₯=30𝑦=βˆ’4,;,
  • Dπ‘₯=20𝑦=βˆ’6π‘₯=6𝑦=20,;,
  • Eπ‘₯=86𝑦=60π‘₯=18𝑦=βˆ’8,;,

Q7:

Solve the simultaneous equations 𝑦=π‘₯βˆ’4,π‘₯5+𝑦3=4, giving your answers to two decimal places.

  • Aπ‘₯=4.46 and 𝑦=0.46, π‘₯=2.46 and 𝑦=1.54
  • Bπ‘₯=0.56 and 𝑦=βˆ’3.44, π‘₯=4.44 and 𝑦=0.44
  • Cπ‘₯=βˆ’0.44 and 𝑦=βˆ’4.44, π‘₯=3.44 and 𝑦=βˆ’0.56
  • Dπ‘₯=6.47 and 𝑦=2.47, π‘₯=βˆ’1.47 and 𝑦=βˆ’5.47
  • Eπ‘₯=βˆ’3.10 and 𝑦=βˆ’7.10, π‘₯=7.10 and 𝑦=3.10

Q8:

Find all of the solutions to the simultaneous equations π‘₯βˆ’π‘¦=6 and π‘₯βˆ’9π‘₯𝑦+𝑦=36.

  • Aπ‘₯=βˆ’12, 𝑦=0; π‘₯=13, 𝑦=6
  • Bπ‘₯=βˆ’6, 𝑦=βˆ’63; π‘₯=0, 𝑦=87
  • Cπ‘₯=0, 𝑦=βˆ’6; π‘₯=6, 𝑦=0
  • Dπ‘₯=βˆ’6, 𝑦=βˆ’63; π‘₯=0, 𝑦=9

Q9:

Find the set of points of intersection of the graphs of π‘₯βˆ’π‘¦=0 and 6π‘₯βˆ’π‘¦=45.

  • A{(3,βˆ’3),(βˆ’3,3)}
  • B{(3,3),(βˆ’3,βˆ’3)}
  • C{(9,βˆ’9),(βˆ’9,9)}
  • D{(9,9),(βˆ’9,βˆ’9)}

Q10:

Find the set of points of intersection of the graphs of π‘₯=βˆ’6+𝑦 and π‘₯βˆ’π‘¦=24.

  • A{(βˆ’7,βˆ’1)}
  • B{(βˆ’1,βˆ’7)}
  • C{(βˆ’5,1)}
  • D{(1,βˆ’5)}

Q11:

Find the set of points of intersection of the graphs of π‘₯+𝑦+6=5 and π‘₯βˆ’π‘¦=35.

  • A{(17,16)}
  • B{(βˆ’18,17)}
  • C{(17,βˆ’18)}
  • D{(16,17)}

Q12:

Find the set of points of intersection of the graphs of π‘₯βˆ’π‘¦=βˆ’7 and π‘₯+π‘¦βˆ’2π‘₯𝑦+5𝑦=14.

  • A{(14,7)}
  • B{(7,14)}
  • C{(βˆ’14,βˆ’7)}
  • D{(βˆ’7,βˆ’14)}

Q13:

Find the set of points of intersection of the graphs of π‘₯+10𝑦=5 and 100π‘¦βˆ’π‘₯=75.

  • A{(βˆ’5,1)}
  • B{(15,1)}
  • C{(1,βˆ’5)}
  • D{(1,15)}

Q14:

Given that 𝑦=4βˆ’π‘₯ and (π‘₯+𝑦)+𝑦=1, find 𝑦.

Q15:

Solve the simultaneous equations 𝑦=π‘₯βˆ’2,(π‘₯βˆ’2)+(π‘¦βˆ’3)=9.

  • Aπ‘₯=1 and 𝑦=0, π‘₯=5 and 𝑦=3
  • Bπ‘₯=2 and 𝑦=0, π‘₯=5 and 𝑦=3
  • Cπ‘₯=4 and 𝑦=2, π‘₯=5 and 𝑦=3
  • Dπ‘₯=0 and 𝑦=βˆ’3, π‘₯=4 and 𝑦=5
  • Eπ‘₯=βˆ’1 and 𝑦=βˆ’3, π‘₯=5 and 𝑦=3

Q16:

The sum of two numbers is 11 and the sum of their squares is 65. What are the numbers?

  • A7, 4
  • B32, βˆ’21
  • C16, 13
  • D8, 2

Q17:

Two numbers have a difference of 6. If the sum of their squares is 68, what are the numbers?

  • A2,βˆ’8 or βˆ’2,8
  • B2,8 or βˆ’2,βˆ’8
  • C2,4 or βˆ’2,βˆ’4
  • D2,βˆ’4 or βˆ’2,4

Q18:

A father’s age is 10 more than 2 times his son’s age. The sum of the squares of their ages is 4 more than 3 times the product of their ages. What are their ages?

  • Athe father’s age =20years, the son’s age =2years
  • Bthe father’s age =52years, the son’s age =15years
  • Cthe father’s age =52years, the son’s age =16years
  • Dthe father’s age =40years, the son’s age =6years
  • Ethe father’s age =42years, the son’s age =16years

Q19:

A right triangle has a hypotenuse of 35 cm and a perimeter of 84 cm. Find the lengths of the other two sides.

  • A21 cm, 26 cm
  • B28 cm, 21 cm
  • C28 cm, 56 cm
  • D56 cm, 42 cm
  • E42 cm, 28 cm

Q20:

The difference between the perimeters of two square yards is 4 m and the difference between their areas is 39 m2. Find the side length of each yard.

  • A21 metres, 18 metres
  • B20 meters, 19 meters
  • C84 metres, 72 metres
  • D80 metres, 76 metres

Q21:

What is the π‘₯-coordinate of the point where the line 2π‘₯+𝑦=βˆ’11 meets the curve 𝑦=4π‘₯+4π‘₯βˆ’3?

  • Aβˆ’14
  • Bβˆ’34
  • Cβˆ’32
  • Dβˆ’158

Q22:

Find the two points on the line 18π‘₯βˆ’π‘¦=12 with a 𝑦-coordinate which is six times the square of the π‘₯ coordinate.

  • A(4,βˆ’6), (2,6)
  • B(βˆ’4,6), (2,6)
  • C(2,6), (2,24)
  • D(2,24), (1,6)
  • E(βˆ’2,24), (βˆ’1,6)

Q23:

Solve the simultaneous equations 𝑦=3π‘₯,𝑦=π‘₯+2π‘₯βˆ’6.

  • Aπ‘₯=βˆ’3 and 𝑦=βˆ’9, π‘₯=2 and 𝑦=6
  • Bπ‘₯=βˆ’2 and 𝑦=βˆ’6, π‘₯=3 and 𝑦=9
  • Cπ‘₯=1 and 𝑦=3, π‘₯=6 and 𝑦=18
  • Dπ‘₯=2 and 𝑦=6, π‘₯=3 and 𝑦=9
  • Eπ‘₯=βˆ’6 and 𝑦=βˆ’18, π‘₯=1 and 𝑦=3

Q24:

Given that (π‘Ž+6,βˆ’4)=𝑏,βˆ’4π‘ο…οŠ¨, determine the values of π‘Ž and 𝑏.

  • Aπ‘Ž=1, 𝑏=1
  • Bπ‘Ž=7, 𝑏=1
  • Cπ‘Ž=βˆ’5, 𝑏=1
  • Dπ‘Ž=βˆ’5, 𝑏=βˆ’4
  • Eπ‘Ž=1, 𝑏=βˆ’5

Q25:

Given that π‘šπ‘Ž=6 and π‘šπ‘Ž=24, determine the value of π‘Ž.

  • A4
  • B2
  • C2,βˆ’2
  • Dβˆ’4
  • Eβˆ’2

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