Worksheet: Introduction to The System of Linear Equations

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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: 7𝑥−3𝑦+6𝑧=5,5𝑥−2𝑦+2𝑧=11,2𝑥−3𝑦+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,𝑥−5𝑦=21,11𝑥−8𝑦=−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 224−1−1−1256𝑝𝑞𝑟=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 𝐴⎡⎢⎢⎣𝑥𝑥𝑥𝑥⎤⎥⎥⎦=⎡⎢⎢⎢⎣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+𝑥𝑥−𝑥3𝑥+𝑥⎤⎥⎥⎥⎦ in the form 𝐴⎡⎢⎢⎣𝑥𝑥𝑥𝑥⎤⎥⎥⎦, 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 ⎡⎢⎢⎢⎣𝑥−𝑥+2𝑥2𝑥+𝑥3𝑥3𝑥+3𝑥+𝑥⎤⎥⎥⎥⎦ in the form 𝐴⎡⎢⎢⎣𝑥𝑥𝑥𝑥⎤⎥⎥⎦, 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 4−237𝑝𝑞=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. 1−2−410134−8𝑝𝑞𝑟=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 −14−2−2153−1−2𝑎𝑏𝑐=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 7𝑥−3𝑦=5,5𝑥−2𝑦=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 13𝑥−23𝑦=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. 3𝑥−5𝑦=24,−9𝑥+7𝑦=20,−2𝑥−8𝑦=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. 11−394𝑥𝑦=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 3𝑥−24=−8𝑦𝑥=3−𝑦 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 4𝑥−2𝑦=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 2𝑥−3𝑦=1 and −2𝑥−3𝑦=−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. 35−12𝑥𝑦=23−4

  • 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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