Question

3. Solve the following system of equations.
2x +y +z = 11
3x + 5y + 5z = 34
5x + 6y + 3z = 42
Part A (10 points) Solve using Gaussian
elimination or a matrix. If you use a
matrix, you must write out your original
matrix and answer matrix. Show your
answer as a point (x, y, z)
I
Part B (2 points) Show that your
solution works in at least one of the
original equations.

Answer 1

The required solution to the given system of equations is (3, 4, 1)

a) Given the systems of equations

2x +y +z = 11 .............................. 1

3x + 5y + 5z = 34 .......................2

5x + 6y + 3z = 42 ..........................3

Reduce the given equation to two equations with two unknown:

Multiply equation 1 by 5 and 2 by 1 to have;

2x +y +z = 11 .............................. 1 × 5

3x + 5y + 5z = 34 .......................2 ×1

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10x +5y +5z = 55 .............................. 4

3x + 5y + 5z = 34 .......................5

Subtract equation 4 from 5:

10x - 3x = 55 - 34

7x = 21

x = 21/7

x = 3

Substitue x = 3 into equation 2 and 3 to have:

3(3) + 5y + 5z = 34 .......................6

5(3) + 6y + 3z = 42........................7

_________________________________________

5y + 5z = 25

6y + 3z = 27

The resulting equations can be reduced to;

y + z = 5

2y + z = 9

Subtracting both expressions

y - 2y = 5 - 9

-y = - 4

y = 4

Substitute x = 3 and y = 4 into equation 1;

2x +y +z = 11

2(3) + 4 + z = 11

6 + 4 + z = 11

z = 11-10

z = 1

Hence the required solution to the given system of equations is (3, 4, 1)

b) Check:

Substitute the solutions into equation 1;

2x +y +z = 11

2(3) +4 + 1

6 + 4 + 1 = 11

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