Worksheet: Solving a System of Two Equations Using a Matrix Inverse

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In this worksheet, we will practice solving a system of two linear equations using the inverse of the matrix of coefficients.

Q1:

Use the inverse matrix to solve the system 3βˆ’154π‘₯π‘¦οŸ=1734, giving your answer as an appropriate matrix.

  • A10217
  • B16
  • Cο”βˆ’6βˆ’1
  • D61
  • E6βˆ’5

Q2:

Given that 581βˆ’8π‘₯π‘¦οŸ=ο”βˆ’431, determine the values of π‘₯ and 𝑦.

  • Aπ‘₯=βˆ’43, 𝑦=1
  • Bπ‘₯=13, 𝑦=βˆ’7
  • Cπ‘₯=βˆ’1, 𝑦=βˆ’7
  • Dπ‘₯=βˆ’7, 𝑦=βˆ’1

Q3:

Given that ο”βˆ’11βˆ’1βˆ’1π‘₯π‘¦οŸ=ο”βˆ’75, find π‘₯π‘¦οŸ.

  • Aο”βˆ’61
  • Bο”βˆ’73
  • C3βˆ’7
  • D1βˆ’6

Q4:

Given that ο”βˆ’3βˆ’52βˆ’8π‘₯π‘¦οŸ=ο”βˆ’64, find π‘₯π‘¦οŸ.

  • A02
  • Bο”βˆ’2βˆ’14
  • Cο”βˆ’14βˆ’2
  • D20

Q5:

Given that 𝐴=2βˆ’5βˆ’8βˆ’9,𝐴π‘₯π‘¦οŸ=ο”βˆ’28, what is 𝑦?

Q6:

Consider the matrices 𝐴=ο”βˆ’4βˆ’24βˆ’8,𝐡=ο”βˆ’848βˆ’4.

If 𝐴×𝐢=𝐡, determine 𝐢.

  • A2βˆ’100
  • Bο”βˆ’2100
  • C1403
  • D1001

Q7:

Given that 1βˆ’5βˆ’45π‘₯π‘¦οŸ=ο”βˆ’2510, determine the values of π‘₯ and 𝑦.

  • Aπ‘₯=βˆ’25, 𝑦=10
  • Bπ‘₯=βˆ’4, 𝑦=1
  • Cπ‘₯=6, 𝑦=5
  • Dπ‘₯=5, 𝑦=6

Q8:

Given that 10792π‘₯π‘¦οŸ=3334, determine the values of π‘₯ and 𝑦.

  • Aπ‘₯=33, 𝑦=34
  • Bπ‘₯=17, 𝑦=11
  • Cπ‘₯=βˆ’1, 𝑦=4
  • Dπ‘₯=4, 𝑦=βˆ’1

Q9:

Given that ο”βˆ’8βˆ’636π‘₯π‘¦οŸ=ο”βˆ’4424, determine the values of π‘₯ and 𝑦.

  • Aπ‘₯=βˆ’44, 𝑦=24
  • Bπ‘₯=βˆ’14, 𝑦=9
  • Cπ‘₯=2, 𝑦=4
  • Dπ‘₯=4, 𝑦=2

Q10:

Given that ο”βˆ’19βˆ’32π‘₯π‘¦οŸ=ο”βˆ’316, determine the values of π‘₯ and 𝑦.

  • Aπ‘₯=βˆ’3, 𝑦=16
  • Bπ‘₯=8, 𝑦=βˆ’1
  • Cπ‘₯=βˆ’1, 𝑦=βˆ’6
  • Dπ‘₯=βˆ’6, 𝑦=βˆ’1

Q11:

Given that 336βˆ’3π‘₯π‘¦οŸ=459, determine the values of π‘₯ and 𝑦.

  • Aπ‘₯=45, 𝑦=9
  • Bπ‘₯=6, 𝑦=3
  • Cπ‘₯=9, 𝑦=6
  • Dπ‘₯=6, 𝑦=9

Q12:

Given that ο”βˆ’7βˆ’1βˆ’4βˆ’10π‘₯π‘¦οŸ=37106, determine the values of π‘₯ and 𝑦.

  • Aπ‘₯=37, 𝑦=106
  • Bπ‘₯=βˆ’8, 𝑦=βˆ’14
  • Cπ‘₯=βˆ’9, 𝑦=βˆ’4
  • Dπ‘₯=βˆ’4, 𝑦=βˆ’9

Q13:

Use matrices to solve the system of equations 3π‘₯+4𝑦=20,2π‘₯+2𝑦=12.

  • Aπ‘₯π‘¦οŸ=438
  • Bπ‘₯π‘¦οŸ=28βˆ’18
  • Cπ‘₯π‘¦οŸ=ο”βˆ’5636
  • Dπ‘₯π‘¦οŸ=ο”βˆ’84
  • Eπ‘₯π‘¦οŸ=42

Q14:

Use matrices to solve the system βˆ’π‘₯+5𝑦=8,βˆ’3π‘₯+𝑦=8.

  • Aπ‘₯=167, 𝑦=βˆ’87
  • Bπ‘₯=5, 𝑦=135
  • Cπ‘₯=βˆ’167, 𝑦=87
  • Dπ‘₯=2, 𝑦=βˆ’1
  • Eπ‘₯=87, 𝑦=βˆ’167

Q15:

Use matrices to solve the systemπ‘₯=9βˆ’5𝑦,8𝑦=9βˆ’7π‘₯.

  • Aπ‘₯=2, 𝑦=βˆ’1
  • Bπ‘₯=1, 𝑦=βˆ’2
  • Cπ‘₯=βˆ’1, 𝑦=2
  • Dπ‘₯=2743, 𝑦=βˆ’5443
  • Eπ‘₯=10, 𝑦=βˆ’15

Q16:

Use matrices to solve the system of equations 4π‘₯βˆ’5𝑦=22,2π‘₯+𝑦=18.

  • Aπ‘₯π‘¦οŸ=82
  • Bπ‘₯π‘¦οŸ=93
  • Cπ‘₯π‘¦οŸ=218
  • Dπ‘₯π‘¦οŸ=11228
  • Eπ‘₯π‘¦οŸ=184

Q17:

Use matrices to solve the system of equations 𝑛+1=2π‘š,𝑛=π‘š+2.

  • Aο“π‘šπ‘›οŸ=24
  • Bο“π‘šπ‘›οŸ=13
  • Cο“π‘šπ‘›οŸ=ο”βˆ’11
  • Dο“π‘šπ‘›οŸ=ο”βˆ’20
  • Eο“π‘šπ‘›οŸ=35

Q18:

Consider the simultaneous equations 4π‘₯βˆ’2𝑦=0,3𝑦+5π‘₯=βˆ’11.

Express the given simultaneous equations as a matrix equation.

  • A4βˆ’253π‘₯π‘¦οŸ=ο”βˆ’110
  • B4βˆ’235π‘₯π‘¦οŸ=0βˆ’11
  • C4βˆ’253π‘₯π‘¦οŸ=0βˆ’11
  • D4βˆ’235π‘₯π‘¦οŸ=ο”βˆ’110
  • E43βˆ’25π‘₯π‘¦οŸ=0βˆ’11

Write down the inverse of the coefficient matrix.

  • A11452βˆ’34
  • B12232βˆ’54
  • C12652βˆ’34
  • D1232βˆ’54
  • E12632βˆ’54

Multiply through by the inverse, on the left-hand side, to solve the matrix equation.

  • Aπ‘₯π‘¦οŸ=1βˆ’1
  • Bπ‘₯π‘¦οŸ=21
  • Cπ‘₯π‘¦οŸ=12
  • Dπ‘₯π‘¦οŸ=ο”βˆ’1βˆ’2
  • Eπ‘₯π‘¦οŸ=ο”βˆ’1βˆ’3

Q19:

Use matrices to solve the following system of equations: 4π‘₯+𝑦=33,3π‘₯+4𝑦=28.

  • Aπ‘₯π‘¦οŸ=51
  • Bπ‘₯π‘¦οŸ=7948
  • Cπ‘₯π‘¦οŸ=63
  • Dπ‘₯π‘¦οŸ=81
  • Eπ‘₯π‘¦οŸ=42

Q20:

Half the difference between two numbers is 1, and the sum of the greater number and double the smaller number is 20. Use matrices to find the two numbers.

  • AThe numbers are 8 and 6.
  • BThe numbers are 24 and 18.
  • CThe numbers are 16 and 2.
  • DThe numbers are 10 and 11.
  • EThe numbers are βˆ’8 and βˆ’6.

Q21:

The straight line whose equation is 𝑦+π‘Žπ‘₯=𝑐 passes through the two points (βˆ’5,5) and (3,10). Using matrices, find π‘Ž and 𝑐.

  • Aπ‘Ž=βˆ’58, 𝑐=658
  • Bπ‘Ž=βˆ’158, 𝑐=358
  • Cπ‘Ž=3, 𝑐=19
  • Dπ‘Ž=10, 𝑐=βˆ’45
  • Eπ‘Ž=5, 𝑐=βˆ’65

Q22:

The length of a rectangle is 6 cm more than twice its width, and twice its length is 39 cm more than its width. Given this, use matrices to determine the perimeter of the rectangle.

Q23:

Madison and Hannah went to Cairo International Book Fair. Madison bought 10 science books and 7 history books and paid 401 LE, while Hannah bought 10 science books and 8 history books and paid 434 LE. Given that all the history books had the same price and all the science books had the same price, use matrices to find the price of a science book and the price of a history book.

  • AEach science book costs 33 LE and each history book costs 17 LE.
  • BEach science book costs 7.40 LE and each history book costs 45 LE.
  • CEach science book costs 14 LE and each history book costs 37.30 LE.
  • DEach science book costs 17 LE and each history book costs 33 LE.

Q24:

A girl bought 37 kilograms of flour and 4 kilograms of butter for 340 LE, and her friend bought 13 kg of flour and 12 kg of butter for 236 LE. Using matrices, find the price per kilogram of both flour and butter.

  • AA kilogram of flour costs 22 LE and a kilogram of butter costs 43 LE.
  • BA kilogram of flour costs 11 LE and a kilogram of butter costs 8 LE.
  • CA kilogram of flour costs 35 LE and a kilogram of butter costs 239 LE.
  • DA kilogram of flour costs 8 LE and a kilogram of butter costs 11 LE.

Q25:

Use matrices to find the two numbers whose sum is 8 and whose difference is 10.

  • Aβˆ’9, 1
  • Bβˆ’18, 2
  • C9, βˆ’1
  • D18, βˆ’2

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