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 2 π‘₯  + π‘₯ + 3 𝑦 = 2 1 .

  • A π‘₯ = 1 , 𝑦 = 5 ; π‘₯ = 0 , 𝑦 = 6
  • B π‘₯ = 1 , 𝑦 = 5 ; π‘₯ = 0 , 𝑦 = 7
  • C π‘₯ = βˆ’ 2 , 𝑦 = 6 ; π‘₯ = 1 , 𝑦 = 7
  • D π‘₯ = 1 , 𝑦 = 6 ; π‘₯ = 0 , 𝑦 = 7

Q2:

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

  • A { ( 0 , 0 ) , ( 5 , βˆ’ 2 5 ) }
  • B { ( 0 , 0 ) , ( βˆ’ 2 5 , βˆ’ 5 ) }
  • C { ( 0 , 0 ) , ( βˆ’ 5 , βˆ’ 2 5 ) }
  • D { ( 0 , 0 ) , ( βˆ’ 2 5 , 5 ) }

Q3:

Given that π‘₯  + π‘₯ 𝑦 = 1 8 , 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  = 5 0 .

  • A { ( 7 , 1 5 ) , ( 1 , 9 ) }
  • B { ( 7 , 7 ) , ( 1 , 1 ) }
  • C { ( 1 5 , 7 ) , ( 9 , 1 ) }
  • D { ( 7 , 1 ) , ( 1 , 7 ) }

Q5:

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

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

Q6:

Find all of the solutions to the simultaneous equations π‘₯ βˆ’ 𝑦 = 2 6 and π‘₯ 𝑦 = 1 2 0 .

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

Q7:

Solve the simultaneous equations giving your answers to two decimal places.

  • A π‘₯ = βˆ’ 0 . 4 4 and 𝑦 = βˆ’ 4 . 4 4 , π‘₯ = 3 . 4 4 and 𝑦 = βˆ’ 0 . 5 6
  • B π‘₯ = βˆ’ 3 . 1 0 and 𝑦 = βˆ’ 7 . 1 0 , π‘₯ = 7 . 1 0 and 𝑦 = 3 . 1 0
  • C π‘₯ = 6 . 4 7 and 𝑦 = 2 . 4 7 , π‘₯ = βˆ’ 1 . 4 7 and 𝑦 = βˆ’ 5 . 4 7
  • D π‘₯ = 0 . 5 6 and 𝑦 = βˆ’ 3 . 4 4 , π‘₯ = 4 . 4 4 and 𝑦 = 0 . 4 4
  • E π‘₯ = 4 . 4 6 and 𝑦 = 0 . 4 6 , π‘₯ = 2 . 4 6 and 𝑦 = 1 . 5 4

Q8:

Find all of the solutions to the simultaneous equations π‘₯ βˆ’ 𝑦 = 6 and π‘₯ βˆ’ 9 π‘₯ 𝑦 + 𝑦 = 3 6   .

  • A π‘₯ = βˆ’ 6 𝑦 = βˆ’ 6 3 π‘₯ = 0 𝑦 = 8 7 , ; ,
  • B π‘₯ = βˆ’ 6 𝑦 = βˆ’ 6 3 π‘₯ = 0 𝑦 = 9 , ; ,
  • C π‘₯ = βˆ’ 1 2 𝑦 = 0 π‘₯ = 1 3 𝑦 = 6 , ; ,
  • D π‘₯ = 0 𝑦 = βˆ’ 6 π‘₯ = 6 𝑦 = 0 , ; ,

Q9:

Find the set of points of intersection of the graphs of π‘₯ βˆ’ 𝑦 = 0 and 6 π‘₯ βˆ’ 𝑦 = 4 5   .

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

Q10:

Find the set of points of intersection of the graphs of π‘₯ = βˆ’ 6 + 𝑦 and π‘₯  βˆ’ 𝑦  = 2 4 .

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

Q11:

Find the set of points of intersection of the graphs of π‘₯ + 𝑦 + 6 = 5 and π‘₯ βˆ’ 𝑦 = 3 5   .

  • A { ( 1 6 , 1 7 ) }
  • B { ( 1 7 , βˆ’ 1 8 ) }
  • C { ( 1 7 , 1 6 ) }
  • D { ( βˆ’ 1 8 , 1 7 ) }

Q12:

Find the set of points of intersection of the graphs of π‘₯ βˆ’ 𝑦 = βˆ’ 7 and π‘₯ + 𝑦 βˆ’ 2 π‘₯ 𝑦 + 5 𝑦 = 1 4   .

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

Q13:

Find the set of points of intersection of the graphs of π‘₯ + 1 0 𝑦 = 5 and 1 0 0 𝑦 βˆ’ π‘₯ = 7 5   .

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

Q14:

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

Q15:

Solve the given simultaneous equations.

  • A π‘₯ = 0 and 𝑦 = βˆ’ 3 , π‘₯ = 4 and 𝑦 = 5
  • B π‘₯ = βˆ’ 1 and 𝑦 = βˆ’ 3 , π‘₯ = 5 and 𝑦 = 3
  • C π‘₯ = 4 and 𝑦 = 2 , π‘₯ = 5 and 𝑦 = 3
  • D π‘₯ = 2 and 𝑦 = 0 , π‘₯ = 5 and 𝑦 = 3
  • E π‘₯ = 1 and 𝑦 = 0 , 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 8 , 2
  • B 1 6 , 1 3
  • C 3 2 , βˆ’ 2 1
  • D 7 , 4

Q17:

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

  • A 2 , βˆ’ 4 or βˆ’ 2 , 4
  • B 2 , βˆ’ 8 or βˆ’ 2 , 8
  • C 2 , 4 or βˆ’ 2 , βˆ’ 4
  • D 2 , 8 or βˆ’ 2 , βˆ’ 8

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?

  • A the father’s age = 5 2 y e a r s , the son’s age = 1 6 y e a r s
  • B the father’s age = 4 0 y e a r s , the son’s age = 6 y e a r s
  • C the father’s age = 2 0 y e a r s , the son’s age = 2 y e a r s
  • D the father’s age = 4 2 y e a r s , the son’s age = 1 6 y e a r s
  • E the father’s age = 5 2 y e a r s , the son’s age = 1 5 y e a r s

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

Q20:

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

  • A80 metres, 76 metres
  • B21 metres, 18 metres
  • C84 metres, 72 metres
  • D20 metres, 19 metres

Q21:

What is the π‘₯ -coordinate of the point where the line 2 π‘₯ + 𝑦 = βˆ’ 1 1 meets the curve 𝑦 = 4 π‘₯  + 4 π‘₯ βˆ’ 3 ?

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

Q22:

Find the two points on the line 1 8 π‘₯ βˆ’ 𝑦 = 1 2 with a 𝑦 -coordinate which is six times the square of the π‘₯ coordinate.

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

Q23:

Solve the simultaneous equations

  • A π‘₯ = βˆ’ 3 and 𝑦 = βˆ’ 9 , π‘₯ = 2 and 𝑦 = 6
  • B π‘₯ = βˆ’ 6 and 𝑦 = βˆ’ 1 8 , π‘₯ = 1 and 𝑦 = 3
  • C π‘₯ = 1 and 𝑦 = 3 , π‘₯ = 6 and 𝑦 = 1 8
  • D π‘₯ = βˆ’ 2 and 𝑦 = βˆ’ 6 , π‘₯ = 3 and 𝑦 = 9
  • E π‘₯ = 2 and 𝑦 = 6 , π‘₯ = 3 and 𝑦 = 9

Q24:

Given that ( π‘Ž + 6 , βˆ’ 4 ) = ο€Ή 𝑏 , βˆ’ 4 𝑏   , determine the values of π‘Ž and 𝑏 .

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

Q25:

Given that π‘š π‘Ž = 6 and π‘š π‘Ž = 2 4 2 , determine the value of π‘Ž .

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

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