A three-variable linear equations are in the form:

*ax + by +cz = p**dx +ey + fz = q**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 *

- = (2.16, -12.21, 2.1)
- = (2.7, 24, 28.4)
- = (5.4, 2.7, -22.2)
- = (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 *

- null
- (-2.1, 15, 3.8)
- (-4.2, -2.1, 0.3)
- (-2.1, 0.3, 1.9)
- (-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*

- (-1.125, 1.375, 0)
- (-1.125, 1.375, 5)
- (4, 4, 10)
- (8, 4, 3.5)

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

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

- (-36, -18, 20)
- (-18, 20, -2)
- (-18, 20, 3.5)
- (2, 8, 5)

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

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

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

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