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A three-variable linear equations are in the form:

1. ax + by +cz = p
2. dx +ey + fz = q
3. gx + hy + iz = r

Just like two variable linear equations, the easiest method for solving these equations is using elimination method.

Let’s solve an example for better understanding of the method.

Example: Consider the set of equations:

5x + 6y + 3z = 1 … (1)
8x + 2y + 3z = 6 … (2)
5x + 2y + 2z = 13 … (3)

Solution:

Subtracting eq (2) from eq (1) we obtain equation (4):

5x + 6y + 3z = 1
-8x – 2y – 3z = -6

-3x + 4y = -5 … (4)

Multiplying equation 2 with 2 & equation 3 by 3 and the subtracting 3(eq3) from 2(eq2) we obtain equation (5):

16x + 4y + 6z = 12 … 2(Eq2)
15x + 6y + 6z = 39 … 3(Eq3)

1x -2y = -27 … (5)

Multiplying equation (5) by 2 and then adding equation 4 and 2(Eq5) we obntain:

-3x + 4y = -5
2x + -4y = -54

-x = -59

or x = 59

Putting value of x in equation (4) we obtain y = 43

Putting value of x and y in eq (1) we obtain z= -184

The MCQs given below contain a set of three different equations. For each question a possible four possible sets of answers is provided. Find the correct answer.

10x + 3y + 3z = 3
2x + 5y + 8z = 8
8x + 5y + 7z = 10

1. = (2.16, -12.21, 2.1)
2. = (2.7, 24, 28.4)
3. = (5.4, 2.7, -22.2)
4. = (2.7, -22.2, 14.2)

Correct answer: 4. (2.7, -22.2, 14.2)

4x + 7y + 7z = 7
1x + 2y + 5z = 8
5x + 5y + 10z = 10

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1. (-2.1, 15, 3.8)
2. (-4.2, -2.1, 0.3)
3. (-2.1, 0.3, 1.9)
4. (-2.1, 0.3, 5)

Correct answer: 3. (-2.1, 0.3, 1.9)

4x + 4y + 7z = 1
6x + 10y + 2z = 7
4x + 4y + 10z = 1

1. (-1.125, 1.375, 0)
2. (-1.125, 1.375, 5)
3. (4, 4, 10)
4. (8, 4, 3.5)

Correct answer: 1. (-1.125, 1.375, 0)

2x + 3y + 9z = 6
5x + 5y + 3z = 4
8x + 8y + 7z = 2

1. (-36, -18, 20)
2. (-18, 20, -2)
3. (-18, 20, 3.5)
4. (2, 8, 5)

Correct answer: 3. (-18, 20, 3.5)

2x + 1y + 7z = 6
6x + 10y + 10z = 7
7x + 10y + 10z = 5

1. (-1.6, 0.3025, 0.7)
2. (-2, 30, 2.7)
3. (-4, -2, 0.55)
4. (-2, 0.55, 1.35)

Correct answer: 4. (-2, 0.55, 1.35)