Worksheet: Introduction to The System of Linear Equations

In this worksheet, we will practice geometrically visualizing a system of equations and expressing it as a matrix equation.

Q1:

Express the following set of simultaneous equations as a matrix equation: 7𝑥3𝑦+6𝑧=5,5𝑥2𝑦+2𝑧=11,2𝑥3𝑦+8𝑧=10.

  • A 7 5 2 3 2 3 6 2 8 𝑥 𝑦 𝑧 = 5 1 1 1 0
  • B 7 5 2 3 2 3 6 2 8 𝑥 𝑦 𝑧 = 1 0 5 1 1
  • C 7 3 6 5 2 2 2 3 8 𝑥 𝑦 𝑧 = 1 0 5 1 1
  • D 6 3 7 2 2 5 8 3 2 𝑥 𝑦 𝑧 = 5 1 1 1 0
  • E 7 3 6 5 2 2 2 3 8 𝑥 𝑦 𝑧 = 5 1 1 1 0

Q2:

Express the set of simultaneous equations 3𝑥=12+5𝑦+2𝑧,𝑥5𝑦=21,11𝑥8𝑦=10+2𝑧 as a matrix equation.

  • A 3 1 1 1 5 5 8 2 0 2 𝑥 𝑦 𝑧 = 1 2 2 1 1 0
  • B 3 5 2 1 5 0 1 1 8 2 𝑥 𝑦 𝑧 = 2 1 1 0 1 2
  • C 3 1 1 1 5 5 8 2 0 2 𝑥 𝑦 𝑧 = 2 1 1 0 1 2
  • D 3 5 2 1 5 0 1 1 8 2 𝑥 𝑦 𝑧 = 1 2 2 1 1 0
  • E 3 5 2 1 5 0 1 1 8 2 𝑥 𝑦 𝑧 = 1 2 2 1 1 0

Q3:

Write down the set of simultaneous equations that could be solved using the matrix equation 224111256𝑝𝑞𝑟=41410.

  • A 2 𝑝 𝑞 + 2 𝑟 = 1 4 , 2 𝑝 𝑞 + 5 𝑟 = 1 0 , 4 𝑝 𝑞 + 6 𝑟 = 4
  • B 2 𝑝 + 4 𝑞 + 2 𝑟 = 4 , 𝑝 𝑞 𝑟 = 1 4 , 2 𝑝 + 6 𝑞 + 5 𝑟 = 1 0
  • C 2 𝑝 𝑞 + 2 𝑟 = 4 , 2 𝑝 𝑞 + 5 𝑟 = 1 4 , 4 𝑝 𝑞 + 6 𝑟 = 1 0
  • D 2 𝑝 + 2 𝑞 + 4 𝑟 = 1 4 , 𝑝 𝑞 𝑟 = 1 0 , 2 𝑝 + 5 𝑞 + 6 𝑟 = 4
  • E 2 𝑝 + 2 𝑞 + 4 𝑟 = 4 , 𝑝 𝑞 𝑟 = 1 4 , 2 𝑝 + 5 𝑞 + 6 𝑟 = 1 0

Q4:

Express the simultaneous equations 3𝑎+2𝑏=13,2𝑎+3𝑏=7 as a matrix equation.

  • A 2 3 3 2 𝑎 𝑏 = 1 3 7
  • B 3 2 2 3 𝑎 𝑏 = 1 3 7
  • C 3 2 2 3 𝑎 𝑏 = 7 1 3
  • D 3 3 2 2 𝑎 𝑏 = 7 1 3
  • E 3 3 2 2 𝑎 𝑏 = 1 3 7

Q5:

Find the matrix 𝐴 such that 𝐴𝑥𝑥𝑥𝑥=𝑥+3𝑥+2𝑥2𝑥+𝑥6𝑥𝑥+3𝑥+𝑥.

  • A 1 1 0 1 3 0 0 3 2 2 6 0 0 0 0 1
  • B 1 2 6 1 3 1 0 3 2 0 0 1 0 0 0 0
  • C 1 3 2 0 2 1 0 0 6 0 0 0 1 3 1 0
  • D 1 2 0 1 3 1 6 3 2 0 0 1 0 0 0 0
  • E 1 3 2 0 1 0 2 0 0 0 6 0 1 3 0 1

Q6:

Write 𝑥+𝑥+𝑥2𝑥+𝑥+𝑥𝑥𝑥3𝑥+𝑥 in the form 𝐴𝑥𝑥𝑥𝑥, where 𝐴 is a matrix.

  • A 1 1 1 0 2 1 1 0 1 1 0 0 3 1 0 0 𝑥 𝑥 𝑥 𝑥
  • B 1 2 1 3 1 1 1 1 1 1 0 0 0 0 0 0 𝑥 𝑥 𝑥 𝑥
  • C 1 1 1 0 1 1 2 0 1 0 1 0 1 0 0 3 𝑥 𝑥 𝑥 𝑥
  • D 1 1 1 1 1 1 0 0 1 2 1 0 0 0 0 3 𝑥 𝑥 𝑥 𝑥
  • E 1 2 1 3 1 1 1 1 1 1 0 1 0 0 0 1 𝑥 𝑥 𝑥 𝑥

Q7:

Write 𝑥𝑥+2𝑥2𝑥+𝑥3𝑥3𝑥+3𝑥+𝑥 in the form 𝐴𝑥𝑥𝑥𝑥, where 𝐴 is a matrix.

  • A 1 1 2 0 1 0 2 0 0 0 3 0 1 3 0 3 𝑥 𝑥 𝑥 𝑥
  • B 1 1 2 0 2 1 0 0 3 0 0 0 3 3 1 0 𝑥 𝑥 𝑥 𝑥
  • C 1 1 2 0 1 0 2 0 0 0 3 0 1 3 0 3 𝑥 𝑥 𝑥 𝑥
  • D 1 1 0 1 1 0 0 3 2 2 3 0 0 0 0 3 𝑥 𝑥 𝑥 𝑥
  • E 1 2 3 3 1 1 0 3 2 0 0 1 0 0 0 0 𝑥 𝑥 𝑥 𝑥

Q8:

Which of the following shows the set of simultaneous equations that could be solved using the matrix equation 4237𝑝𝑞=414?

  • A 4 𝑝 = 4 + 3 𝑞 , 2 𝑝 + 1 4 = 7 𝑞
  • B 4 𝑝 = 4 2 𝑞 , 3 𝑝 + 1 4 = 7 𝑞
  • C 4 𝑝 = 4 + 2 𝑞 , 3 𝑝 1 4 = 7 𝑞
  • D 4 𝑝 = 4 3 𝑞 , 2 𝑝 1 4 = 7 𝑞
  • E 2 𝑝 = 4 + 4 𝑞 , 7 𝑝 1 4 = 3 𝑞

Q9:

Write down the set of simultaneous equations that could be solved using the given matrix equation. 124101348𝑝𝑞𝑟=11610

  • A 𝑝 2 𝑞 4 𝑟 = 1 1 𝑝 + 𝑞 = 6 3 𝑝 + 4 𝑞 8 𝑟 = 1 0
  • B 𝑝 + 𝑞 + 3 𝑟 = 1 1 2 𝑝 + 4 𝑟 = 6 4 𝑝 + 𝑞 8 𝑟 = 1 0
  • C 𝑝 2 𝑞 4 𝑟 = 6 𝑝 + 𝑟 = 1 0 3 𝑝 + 4 𝑞 8 𝑟 = 1 1
  • D 𝑝 2 𝑞 4 𝑟 = 1 1 𝑝 + 𝑟 = 6 3 𝑝 + 4 𝑞 8 𝑟 = 1 0
  • E 𝑝 + 𝑞 + 3 𝑟 = 6 2 𝑝 + 4 𝑟 = 1 0 4 𝑝 + 𝑞 8 𝑟 = 1 1

Q10:

Which of the following shows the set of simultaneous equations that could be solved using the matrix equation 142215312𝑎𝑏𝑐=5711?

  • A 𝑎 + 4 𝑏 2 𝑐 = 5 2 𝑎 + 𝑏 + 5 𝑐 = 7 3 𝑎 𝑏 2 𝑐 = 1 1
  • B 𝑐 + 4 𝑏 2 𝑎 = 5 2 𝑐 + 𝑏 + 5 𝑎 = 7 3 𝑐 𝑏 2 𝑎 = 1 1
  • C 𝑐 2 𝑏 + 3 𝑎 = 5 4 𝑐 + 𝑏 𝑎 = 7 2 𝑐 + 5 𝑏 2 𝑎 = 1 1
  • D 𝑐 + 4 𝑏 2 𝑎 = 1 1 2 𝑐 + 𝑏 + 5 𝑎 = 7 3 𝑐 𝑏 2 𝑎 = 5
  • E 𝑎 2 𝑏 + 3 𝑐 = 5 4 𝑎 + 𝑏 𝑐 = 7 2 𝑎 + 5 𝑏 2 𝑐 = 1 1

Q11:

Express the simultaneous equations 7𝑥3𝑦=5,5𝑥2𝑦=11 as a matrix equation.

  • A 7 5 3 2 𝑥 𝑦 = 5 1 1
  • B 7 3 5 2 𝑥 𝑦 = 1 1 5
  • C 3 7 2 5 𝑥 𝑦 = 5 1 1
  • D 7 3 5 2 𝑥 𝑦 = 5 1 1
  • E 7 5 3 2 𝑥 𝑦 = 1 1 5

Q12:

Express the simultaneous equations 13𝑥23𝑦=53,34𝑦+14𝑥=74 as a matrix equation.

  • A 1 3 2 3 1 4 3 4 𝑥 𝑦 = 5 3 7 4
  • B 1 3 2 3 3 4 1 4 𝑥 𝑦 = 5 3 7 4
  • C 1 3 3 4 2 3 1 4 𝑥 𝑦 = 7 4 5 3
  • D 1 3 1 4 3 4 2 3 𝑥 𝑦 = 7 4 5 3
  • E 2 3 1 3 3 4 1 4 𝑥 𝑦 = 5 3 7 4

Q13:

Express the following system of equations in matrix form. 3𝑥5𝑦=24,9𝑥+7𝑦=20,2𝑥8𝑦=12.

  • A 3 9 2 5 7 8 𝑥 𝑦 = 2 4 2 0 1 2
  • B 3 5 9 7 2 8 𝑥 𝑦 = 2 4 2 0 1 2
  • C 5 7 8 3 9 2 𝑥 𝑦 = 2 4 2 0 1 2
  • D 5 3 7 9 8 2 𝑥 𝑦 = 2 4 2 0 1 2

Q14:

Write down the set of simultaneous equations that could be solved using the given matrix equation. 11394𝑥𝑦=813

  • A 1 1 𝑥 + 9 𝑦 = 8 3 𝑥 + 4 𝑦 = 1 3
  • B 1 1 𝑥 3 𝑦 = 8 9 𝑥 + 4 𝑦 = 1 3
  • C 1 1 𝑥 3 𝑦 = 1 3 9 𝑥 + 4 𝑦 = 8
  • D 1 1 𝑥 + 9 𝑦 = 1 3 3 𝑥 + 4 𝑦 = 8
  • E 3 𝑥 + 1 1 𝑦 = 8 4 𝑥 + 9 𝑦 = 1 3

Q15:

Express the simultaneous equations 3𝑥24=8𝑦𝑥=3𝑦 as a matrix equation.

  • A 3 8 1 1 𝑥 𝑦 = 2 4 3
  • B 3 1 8 1 𝑥 𝑦 = 2 4 3
  • C 3 8 1 1 𝑥 𝑦 = 2 4 3
  • D 3 1 8 1 𝑥 𝑦 = 3 2 4
  • E 3 8 1 1 𝑥 𝑦 = 3 2 4

Q16:

Express the simultaneous equations 𝑛+1=2𝑚,𝑛=𝑚+2 as a matrix equation.

  • A 2 1 1 1 𝑚 𝑛 = 1 2
  • B 2 1 1 1 𝑚 𝑛 = 1 2
  • C 2 1 1 1 𝑚 𝑛 = 2 1
  • D 2 1 1 1 𝑚 𝑛 = 2 1
  • E 2 1 1 1 𝑚 𝑛 = 1 2

Q17:

Express the simultaneous equations 3𝑥+2𝑦=12,3𝑥+𝑦=7 as a matrix equation.

  • A 3 3 2 1 𝑥 𝑦 = 7 1 2
  • B 3 2 3 1 𝑥 𝑦 = 7 1 2
  • C 3 2 1 3 𝑥 𝑦 = 1 2 7
  • D 3 2 3 1 𝑥 𝑦 = 1 2 7
  • E 3 3 2 1 𝑥 𝑦 = 1 2 7

Q18:

Express the simultaneous equations 4𝑥2𝑦=0,3𝑦+5𝑥=11 as a matrix equation.

  • A 2 4 3 5 𝑥 𝑦 = 1 1 0
  • B 4 2 5 3 𝑥 𝑦 = 0 1 1
  • C 4 2 5 3 𝑥 𝑦 = 1 1 0
  • D 4 3 2 0 𝑥 𝑦 = 0 1 1
  • E 4 3 2 0 𝑥 𝑦 = 1 1 0

Q19:

Which of the following augmented matrices represents the pair of equations 2𝑥3𝑦=1 and 2𝑥3𝑦=1?

  • A 2 2 3 3 1 1
  • B 2 3 1 2 3 1
  • C 2 3 1 2 3 1
  • D 2 2 3 3 1 1
  • E 2 2 3 3 1 1

Q20:

Write down the set of simultaneous equations that could be solved using the given matrix equation. 3512𝑥𝑦=234

  • A 3 𝑦 + 5 𝑥 = 2 3 𝑦 + 2 𝑥 = 4
  • B 3 𝑥 + 5 𝑦 = 2 3 𝑥 + 2 𝑦 = 4
  • C 3 𝑥 𝑦 = 2 3 5 𝑥 + 2 𝑦 = 4
  • D 2 3 𝑥 4 𝑦 = 3 4 𝑥 + 1 7 𝑦 = 1
  • E 3 𝑥 + 5 𝑥 = 2 3 𝑦 + 2 𝑦 = 4

Q21:

Write down the set of simultaneous equations that could be solved using the given matrix equation. 3324𝑎𝑏=1012

  • A 3 𝑎 + 3 𝑏 = 1 0 4 𝑎 + 2 𝑏 = 1 2
  • B 3 𝑎 + 3 𝑏 = 1 0 2 𝑎 + 4 𝑏 = 1 2
  • C 3 𝑎 + 2 𝑏 = 1 0 3 𝑎 + 4 𝑏 = 1 2
  • D 3 𝑎 + 2 𝑏 = 1 2 3 𝑎 + 4 𝑏 = 1 0
  • E 3 𝑎 + 3 𝑏 = 1 2 2 𝑎 + 4 𝑏 = 1 0

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