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.

  • A π‘₯ = 1 , 𝑦 = 5 ; π‘₯ = 0 , 𝑦 = 7
  • B π‘₯ = 1 , 𝑦 = 5 ; π‘₯ = 0 , 𝑦 = 6
  • C π‘₯ = 1 , 𝑦 = 6 ; π‘₯ = 0 , 𝑦 = 7
  • D π‘₯ = 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’ 2 5 ) }
  • B { ( 0 , 0 ) , ( 5 , βˆ’ 2 5 ) }
  • C { ( 0 , 0 ) , ( βˆ’ 2 5 , 5 ) }
  • D { ( 0 , 0 ) , ( βˆ’ 2 5 , 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 { ( 1 5 , 7 ) , ( 9 , 1 ) }
  • C { ( 7 , 1 5 ) , ( 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.

  • A π‘₯ = 2 𝑦 = βˆ’ 5 π‘₯ = 1 𝑦 = 1 , ; ,
  • B π‘₯ = 2 𝑦 = βˆ’ 1 π‘₯ = 1 𝑦 = 1 , ; ,
  • C π‘₯ = βˆ’ 4 𝑦 = βˆ’ 1 π‘₯ = 1 𝑦 = 1 , ; ,
  • D π‘₯ = 2 𝑦 = βˆ’ 5 π‘₯ = 1 𝑦 = βˆ’ 1 , ; ,

Q6:

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

  • A π‘₯ = 3 0 𝑦 = 4 π‘₯ = βˆ’ 4 𝑦 = βˆ’ 3 0 , ; ,
  • B π‘₯ = 4 𝑦 = 6 0 π‘₯ = βˆ’ 3 0 𝑦 = 3 , ; ,
  • C π‘₯ = βˆ’ 4 𝑦 = 3 0 π‘₯ = 3 0 𝑦 = βˆ’ 4 , ; ,
  • D π‘₯ = 2 0 𝑦 = βˆ’ 6 π‘₯ = 6 𝑦 = 2 0 , ; ,
  • E π‘₯ = 8 6 𝑦 = 6 0 π‘₯ = 1 8 𝑦 = βˆ’ 8 , ; ,

Q7:

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

  • A π‘₯ = 4 . 4 6 and 𝑦=0.46, π‘₯=2.46 and 𝑦=1.54
  • B π‘₯ = 0 . 5 6 and 𝑦=βˆ’3.44, π‘₯=4.44 and 𝑦=0.44
  • C π‘₯ = βˆ’ 0 . 4 4 and 𝑦=βˆ’4.44, π‘₯=3.44 and 𝑦=βˆ’0.56
  • D π‘₯ = 6 . 4 7 and 𝑦=2.47, π‘₯=βˆ’1.47 and 𝑦=βˆ’5.47
  • E π‘₯ = βˆ’ 3 . 1 0 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.

  • A π‘₯ = βˆ’ 1 2 𝑦 = 0 π‘₯ = 1 3 𝑦 = 6 , ; ,
  • B π‘₯ = βˆ’ 6 𝑦 = βˆ’ 6 3 π‘₯ = 0 𝑦 = 8 7 , ; ,
  • C π‘₯ = 0 𝑦 = βˆ’ 6 π‘₯ = 6 𝑦 = 0 , ; ,
  • D π‘₯ = βˆ’ 6 𝑦 = βˆ’ 6 3 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 { ( 1 7 , 1 6 ) }
  • B { ( βˆ’ 1 8 , 1 7 ) }
  • C { ( 1 7 , βˆ’ 1 8 ) }
  • D { ( 1 6 , 1 7 ) }

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 { ( 1 4 , 7 ) }
  • B { ( 7 , 1 4 ) }
  • C { ( βˆ’ 1 4 , βˆ’ 7 ) }
  • D { ( βˆ’ 7 , βˆ’ 1 4 ) }

Q13:

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

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

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.(2)

  • A π‘₯ = 1 and 𝑦=0, π‘₯=5 and 𝑦=3
  • B π‘₯ = 2 and 𝑦=0, π‘₯=5 and 𝑦=3
  • C π‘₯ = 4 and 𝑦=2, π‘₯=5 and 𝑦=3
  • D π‘₯ = 0 and 𝑦=βˆ’3, π‘₯=4 and 𝑦=5
  • E π‘₯ = 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?

  • A 7 , 4
  • B 3 2 , βˆ’ 2 1
  • C 1 6 , 1 3
  • D 8 , 2

Q17:

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

  • A 2 , βˆ’ 8 or βˆ’2,8
  • B 2 , 8 or βˆ’2,βˆ’8
  • C 2 , 4 or βˆ’2,βˆ’4
  • D 2 , 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-angled triangle has a hypotenuse of 35 cm and a perimeter of 84 cm. Find the lengths of the other two sides.

  • A 28 cm, 21 cm
  • B 42 cm, 28 cm
  • C 56 cm, 42 cm
  • D 21 cm, 26 cm
  • E 28 cm, 56 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?

  • A βˆ’ 1 4
  • B βˆ’ 3 4
  • C βˆ’ 3 2
  • D βˆ’ 1 5 8

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 , 2 4 )
  • D ( 2 , 2 4 ) , ( 1 , 6 )
  • E ( βˆ’ 2 , 2 4 ) , ( βˆ’ 1 , 6 )

Q23:

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

  • A π‘₯ = βˆ’ 3 and 𝑦=βˆ’9, π‘₯=2 and 𝑦=6
  • B π‘₯ = βˆ’ 2 and 𝑦=βˆ’6, π‘₯=3 and 𝑦=9
  • C π‘₯ = 1 and 𝑦=3, π‘₯=6 and 𝑦=18
  • D π‘₯ = 2 and 𝑦=6, π‘₯=3 and 𝑦=9
  • E π‘₯ = 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.

  • A π‘Ž = 1 , 𝑏 = 1
  • B π‘Ž = 7 , 𝑏 = 1
  • C π‘Ž = βˆ’ 5 , 𝑏 = 1
  • D π‘Ž = βˆ’ 5 , 𝑏 = βˆ’ 4
  • E π‘Ž = 1 , 𝑏 = βˆ’ 5

Q25:

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

  • A 4
  • B 2
  • C 2 , βˆ’ 2
  • D βˆ’ 4
  • E βˆ’ 2

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