Worksheet: Introduction to the System of Linear Equations

In this worksheet, we will practice expressing a system of linear equations as a matrix equation.

Q1:

Express the following set of simultaneous equations as a matrix equation: 7π‘₯βˆ’3𝑦+6𝑧=5,5π‘₯βˆ’2𝑦+2𝑧=11,2π‘₯βˆ’3𝑦+8𝑧=10.

  • A752βˆ’3βˆ’2βˆ’3628π‘₯𝑦𝑧=51110
  • B752βˆ’3βˆ’2βˆ’3628π‘₯𝑦𝑧=10511
  • C7βˆ’365βˆ’22238π‘₯𝑦𝑧=10511
  • D6βˆ’372βˆ’25832π‘₯𝑦𝑧=51110
  • E7βˆ’365βˆ’222βˆ’38π‘₯𝑦𝑧=51110

Q2:

Express the set of simultaneous equations 3π‘₯=12+5𝑦+2𝑧,π‘₯βˆ’5𝑦=21,11π‘₯βˆ’8𝑦=βˆ’10+2𝑧 as a matrix equation.

  • A3111βˆ’5βˆ’5βˆ’8βˆ’20βˆ’2π‘₯𝑦𝑧=1221βˆ’10
  • B3βˆ’5βˆ’21βˆ’5011βˆ’8βˆ’2π‘₯𝑦𝑧=21βˆ’1012
  • C3111βˆ’5βˆ’5βˆ’8βˆ’20βˆ’2π‘₯𝑦𝑧=21βˆ’1012
  • D3βˆ’5βˆ’21βˆ’5011βˆ’8βˆ’2π‘₯𝑦𝑧=1221βˆ’10
  • E3521βˆ’5011βˆ’82π‘₯𝑦𝑧=1221βˆ’10

Q3:

Write down the set of simultaneous equations that could be solved using the matrix equation 224βˆ’1βˆ’1βˆ’1256ο€ο˜π‘π‘žπ‘Ÿο€=41410.

  • A2π‘βˆ’π‘ž+2π‘Ÿ=14, 2π‘βˆ’π‘ž+5π‘Ÿ=10, 4π‘βˆ’π‘ž+6π‘Ÿ=4
  • B2𝑝+4π‘ž+2π‘Ÿ=4, βˆ’π‘βˆ’π‘žβˆ’π‘Ÿ=14, 2𝑝+6π‘ž+5π‘Ÿ=10
  • C2π‘βˆ’π‘ž+2π‘Ÿ=4, 2π‘βˆ’π‘ž+5π‘Ÿ=14, 4π‘βˆ’π‘ž+6π‘Ÿ=10
  • D2𝑝+2π‘ž+4π‘Ÿ=14, βˆ’π‘βˆ’π‘žβˆ’π‘Ÿ=10, 2𝑝+5π‘ž+6π‘Ÿ=4
  • E2𝑝+2π‘ž+4π‘Ÿ=4, βˆ’π‘βˆ’π‘žβˆ’π‘Ÿ=14, 2𝑝+5π‘ž+6π‘Ÿ=10

Q4:

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

  • A2332ο ο“π‘Žπ‘οŸ=137
  • B3223ο ο“π‘Žπ‘οŸ=137
  • C3223ο ο“π‘Žπ‘οŸ=713
  • D3322ο ο“π‘Žπ‘οŸ=713
  • E3322ο ο“π‘Žπ‘οŸ=137

Q5:

Which of the following shows the set of simultaneous equations that could be solved using the matrix equation 4βˆ’237ο ο”π‘π‘žο =414?

  • A4𝑝=4+3π‘ž,βˆ’2𝑝+14=βˆ’7π‘ž
  • B4𝑝=4βˆ’2π‘ž,3𝑝+14=7π‘ž
  • C4𝑝=4+2π‘ž,3π‘βˆ’14=βˆ’7π‘ž
  • D4𝑝=4βˆ’3π‘ž,βˆ’2π‘βˆ’14=7π‘ž
  • E2𝑝=4+4π‘ž,7π‘βˆ’14=βˆ’3π‘ž

Q6:

Write down the set of simultaneous equations that could be solved using the given matrix equation. 1βˆ’2βˆ’410134βˆ’8ο€ο˜π‘π‘žπ‘Ÿο€=11610

  • Aπ‘βˆ’2π‘žβˆ’4π‘Ÿ=11𝑝+π‘ž=63𝑝+4π‘žβˆ’8π‘Ÿ=10
  • B𝑝+π‘ž+3π‘Ÿ=11βˆ’2𝑝+4π‘Ÿ=6βˆ’4𝑝+π‘žβˆ’8π‘Ÿ=10
  • Cπ‘βˆ’2π‘žβˆ’4π‘Ÿ=6𝑝+π‘Ÿ=103𝑝+4π‘žβˆ’8π‘Ÿ=11
  • Dπ‘βˆ’2π‘žβˆ’4π‘Ÿ=11𝑝+π‘Ÿ=63𝑝+4π‘žβˆ’8π‘Ÿ=10
  • E𝑝+π‘ž+3π‘Ÿ=6βˆ’2𝑝+4π‘Ÿ=10βˆ’4𝑝+π‘žβˆ’8π‘Ÿ=11

Q7:

Which of the following shows the set of simultaneous equations that could be solved using the matrix equation ο˜βˆ’14βˆ’2βˆ’2153βˆ’1βˆ’2ο€ο—π‘Žπ‘π‘ο£=5711?

  • Aβˆ’π‘Ž+4π‘βˆ’2𝑐=5βˆ’2π‘Ž+𝑏+5𝑐=73π‘Žβˆ’π‘βˆ’2𝑐=11
  • Bβˆ’π‘+4π‘βˆ’2π‘Ž=5βˆ’2𝑐+𝑏+5π‘Ž=73π‘βˆ’π‘βˆ’2π‘Ž=11
  • Cβˆ’π‘βˆ’2𝑏+3π‘Ž=54𝑐+π‘βˆ’π‘Ž=7βˆ’2𝑐+5π‘βˆ’2π‘Ž=11
  • Dβˆ’π‘+4π‘βˆ’2π‘Ž=11βˆ’2𝑐+𝑏+5π‘Ž=73π‘βˆ’π‘βˆ’2π‘Ž=5
  • Eβˆ’π‘Žβˆ’2𝑏+3𝑐=54π‘Ž+π‘βˆ’π‘=7βˆ’2π‘Ž+5π‘βˆ’2𝑐=11

Q8:

Express the simultaneous equations 7π‘₯βˆ’3𝑦=5,5π‘₯βˆ’2𝑦=11 as a matrix equation.

  • A75βˆ’3βˆ’2π‘₯π‘¦οŸ=511
  • B7βˆ’35βˆ’2π‘₯π‘¦οŸ=115
  • Cο”βˆ’37βˆ’25π‘₯π‘¦οŸ=511
  • D7βˆ’35βˆ’2π‘₯π‘¦οŸ=511
  • E75βˆ’3βˆ’2π‘₯π‘¦οŸ=115

Q9:

Express the simultaneous equations 13π‘₯βˆ’23𝑦=53,34𝑦+14π‘₯=74 as a matrix equation.

  • A⎑⎒⎒⎣13βˆ’231434⎀βŽ₯βŽ₯βŽ¦ο“π‘₯π‘¦οŸ=⎑⎒⎒⎣5374⎀βŽ₯βŽ₯⎦
  • B⎑⎒⎒⎣13βˆ’233414⎀βŽ₯βŽ₯βŽ¦ο“π‘₯π‘¦οŸ=⎑⎒⎒⎣5374⎀βŽ₯βŽ₯⎦
  • C⎑⎒⎒⎣1334βˆ’2314⎀βŽ₯βŽ₯βŽ¦ο“π‘₯π‘¦οŸ=⎑⎒⎒⎣7453⎀βŽ₯βŽ₯⎦
  • D⎑⎒⎒⎣131434βˆ’23⎀βŽ₯βŽ₯βŽ¦ο“π‘₯π‘¦οŸ=⎑⎒⎒⎣7453⎀βŽ₯βŽ₯⎦
  • EβŽ‘βŽ’βŽ’βŽ£βˆ’23133414⎀βŽ₯βŽ₯βŽ¦ο“π‘₯π‘¦οŸ=⎑⎒⎒⎣5374⎀βŽ₯βŽ₯⎦

Q10:

Express the following system of equations in matrix form. 3π‘₯βˆ’5𝑦=24,βˆ’9π‘₯+7𝑦=20,βˆ’2π‘₯βˆ’8𝑦=12.

  • A3βˆ’9βˆ’2βˆ’57βˆ’8π‘₯π‘¦οŸ=242012
  • B3βˆ’5βˆ’97βˆ’2βˆ’8π‘₯π‘¦οŸ=242012
  • Cο”βˆ’57βˆ’83βˆ’9βˆ’2π‘₯π‘¦οŸ=242012
  • Dο˜βˆ’537βˆ’9βˆ’8βˆ’2π‘₯π‘¦οŸ=242012

Q11:

Write down the set of simultaneous equations that could be solved using the given matrix equation. 11βˆ’394π‘₯π‘¦οŸ=813

  • A11π‘₯+9𝑦=8βˆ’3π‘₯+4𝑦=13
  • B11π‘₯βˆ’3𝑦=89π‘₯+4𝑦=13
  • C11π‘₯βˆ’3𝑦=139π‘₯+4𝑦=8
  • D11π‘₯+9𝑦=13βˆ’3π‘₯+4𝑦=8
  • Eβˆ’3π‘₯+11𝑦=84π‘₯+9𝑦=13

Q12:

Express the simultaneous equations 3π‘₯βˆ’24=βˆ’8𝑦π‘₯=3βˆ’π‘¦ as a matrix equation.

  • A3βˆ’81βˆ’1π‘₯π‘¦οŸ=243
  • B3181π‘₯π‘¦οŸ=243
  • C3811π‘₯π‘¦οŸ=243
  • D3181π‘₯π‘¦οŸ=324
  • E3811π‘₯π‘¦οŸ=324

Q13:

Express the simultaneous equations 𝑛+1=2π‘š,𝑛=π‘š+2 as a matrix equation.

  • A2βˆ’11βˆ’1ο ο“π‘šπ‘›οŸ=12
  • B2111ο ο“π‘šπ‘›οŸ=12
  • C2111ο ο“π‘šπ‘›οŸ=21
  • D2βˆ’1βˆ’11ο ο“π‘šπ‘›οŸ=21
  • E2βˆ’1βˆ’11ο ο“π‘šπ‘›οŸ=12

Q14:

Express the simultaneous equations 3π‘₯+2𝑦=12,3π‘₯+𝑦=7 as a matrix equation.

  • A3321π‘₯π‘¦οŸ=712
  • B3231π‘₯π‘¦οŸ=712
  • C3213π‘₯π‘¦οŸ=127
  • D3231π‘₯π‘¦οŸ=127
  • E3321π‘₯π‘¦οŸ=127

Q15:

Express the simultaneous equations 4π‘₯βˆ’2𝑦=0,3𝑦+5π‘₯=βˆ’11 as a matrix equation.

  • Aο”βˆ’2435π‘₯π‘¦οŸ=ο”βˆ’110
  • B4βˆ’253π‘₯π‘¦οŸ=0βˆ’11
  • C4βˆ’253π‘₯π‘¦οŸ=ο”βˆ’110
  • D43βˆ’20π‘₯π‘¦οŸ=0βˆ’11
  • E43βˆ’20π‘₯π‘¦οŸ=ο”βˆ’110

Q16:

Write down the set of simultaneous equations that could be solved using the given matrix equation. 35βˆ’12π‘₯π‘¦οŸ=23βˆ’4

  • A3𝑦+5π‘₯=23βˆ’π‘¦+2π‘₯=βˆ’4
  • B3π‘₯+5𝑦=23βˆ’π‘₯+2𝑦=βˆ’4
  • C3π‘₯βˆ’π‘¦=235π‘₯+2𝑦=βˆ’4
  • D23π‘₯βˆ’4𝑦=3βˆ’4π‘₯+17𝑦=βˆ’1
  • E3π‘₯+5π‘₯=23βˆ’π‘¦+2𝑦=βˆ’4

Q17:

Write down the set of simultaneous equations that could be solved using the given matrix equation. 3324ο ο“π‘Žπ‘οŸ=1012

  • A3π‘Ž+3𝑏=104π‘Ž+2𝑏=12
  • B3π‘Ž+3𝑏=102π‘Ž+4𝑏=12
  • C3π‘Ž+2𝑏=103π‘Ž+4𝑏=12
  • D3π‘Ž+2𝑏=123π‘Ž+4𝑏=10
  • E3π‘Ž+3𝑏=122π‘Ž+4𝑏=10

Q18:

Find the matrix 𝐴 such that 𝐴⎑⎒⎒⎣π‘₯π‘₯π‘₯π‘₯⎀βŽ₯βŽ₯⎦=⎑⎒⎒⎒⎣π‘₯+3π‘₯+2π‘₯2π‘₯+π‘₯6π‘₯π‘₯+3π‘₯+π‘₯⎀βŽ₯βŽ₯βŽ₯⎦.οŠͺοŠͺ

  • A⎑⎒⎒⎣1101300322600001⎀βŽ₯βŽ₯⎦
  • B⎑⎒⎒⎣1261310320010000⎀βŽ₯βŽ₯⎦
  • C⎑⎒⎒⎣1320210060001310⎀βŽ₯βŽ₯⎦
  • D⎑⎒⎒⎣1201316320010000⎀βŽ₯βŽ₯⎦
  • E⎑⎒⎒⎣1320102000601301⎀βŽ₯βŽ₯⎦

Q19:

Write ⎑⎒⎒⎒⎣π‘₯+π‘₯+π‘₯2π‘₯+π‘₯+π‘₯π‘₯βˆ’π‘₯3π‘₯+π‘₯⎀βŽ₯βŽ₯βŽ₯⎦οŠͺ in the form 𝐴⎑⎒⎒⎣π‘₯π‘₯π‘₯π‘₯⎀βŽ₯βŽ₯⎦,οŠͺ where 𝐴 is a matrix.

  • A⎑⎒⎒⎣111021101βˆ’1003100⎀βŽ₯βŽ₯⎦⎑⎒⎒⎣π‘₯π‘₯π‘₯π‘₯⎀βŽ₯βŽ₯⎦οŠͺ
  • B⎑⎒⎒⎣121311βˆ’1111000000⎀βŽ₯βŽ₯⎦⎑⎒⎒⎣π‘₯π‘₯π‘₯π‘₯⎀βŽ₯βŽ₯⎦οŠͺ
  • C⎑⎒⎒⎣11101120βˆ’10101003⎀βŽ₯βŽ₯⎦⎑⎒⎒⎣π‘₯π‘₯π‘₯π‘₯⎀βŽ₯βŽ₯⎦οŠͺ
  • D⎑⎒⎒⎣11βˆ’11110012100003⎀βŽ₯βŽ₯⎦⎑⎒⎒⎣π‘₯π‘₯π‘₯π‘₯⎀βŽ₯βŽ₯⎦οŠͺ
  • E⎑⎒⎒⎣121311βˆ’1111010001⎀βŽ₯βŽ₯⎦⎑⎒⎒⎣π‘₯π‘₯π‘₯π‘₯⎀βŽ₯βŽ₯⎦οŠͺ

Q20:

Write ⎑⎒⎒⎒⎣π‘₯βˆ’π‘₯+2π‘₯2π‘₯+π‘₯3π‘₯3π‘₯+3π‘₯+π‘₯⎀βŽ₯βŽ₯βŽ₯⎦οŠͺ in the form 𝐴⎑⎒⎒⎣π‘₯π‘₯π‘₯π‘₯⎀βŽ₯βŽ₯⎦,οŠͺ where 𝐴 is a matrix.

  • A⎑⎒⎒⎣1120102000301303⎀βŽ₯βŽ₯⎦⎑⎒⎒⎣π‘₯π‘₯π‘₯π‘₯⎀βŽ₯βŽ₯⎦οŠͺ
  • B⎑⎒⎒⎣1βˆ’120210030003310⎀βŽ₯βŽ₯⎦⎑⎒⎒⎣π‘₯π‘₯π‘₯π‘₯⎀βŽ₯βŽ₯⎦οŠͺ
  • C⎑⎒⎒⎣1βˆ’120102000301303⎀βŽ₯βŽ₯⎦⎑⎒⎒⎣π‘₯π‘₯π‘₯π‘₯⎀βŽ₯βŽ₯⎦οŠͺ
  • D⎑⎒⎒⎣1101βˆ’100322300003⎀βŽ₯βŽ₯⎦⎑⎒⎒⎣π‘₯π‘₯π‘₯π‘₯⎀βŽ₯βŽ₯⎦οŠͺ
  • E⎑⎒⎒⎣1233βˆ’110320010000⎀βŽ₯βŽ₯⎦⎑⎒⎒⎣π‘₯π‘₯π‘₯π‘₯⎀βŽ₯βŽ₯⎦οŠͺ

Q21:

True or False: In the matrix equation 𝐴𝑋=𝐡, 𝐴 must be a square matrix.

  • AFalse
  • BTrue

Q22:

True or False: The number of columns of a coefficient matrix in a matrix equation represents the number of variables.

  • ATrue
  • BFalse

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