GATE CS 2014 Set 1 Q.14 (CS Q.4): Linear Algebra

Consider the following system of equations:

3x + 2y = 1

4x + 7z = 1

x + y + z = 3

x - 2y + 7z = 0

The number of solutions for this system is __________________

Your answer

Official answer and solution

Official answer: 1
From: Official GATE 2014 answer key

Solution

Solve the linear system by substitution and Gaussian elimination to determine consistency and the number of solutions.

The system consists of 4 equations in 3 variables ( $x, y, z$ ):

$3x + 2y = 1$
$4x + 7z = 1$
$x + y + z = 3$
$x - 2y + 7z = 0$

From equation (3):

y = 3 - x - z

Substitute $y$ into equation (1):

3x + 2(3 - x - z) = 1 \implies 3x + 6 - 2x - 2z = 1 \implies x - 2z = -5 \implies x = 2z - 5

Substitute $x$ into equation (2):

4(2z - 5) + 7z = 1 \implies 8z - 20 + 7z = 1 \implies 15z = 21 \implies z = \frac{21}{15} = \frac{7}{5} = 1.4

Now find $x$ and $y$ :

x = 2(1.4) - 5 = 2.8 - 5 = -2.2

y = 3 - (-2.2) - 1.4 = 3 + 2.2 - 1.4 = 3.8

Now verify if these values satisfy equation (4):

x - 2y + 7z = -2.2 - 2(3.8) + 7(1.4) = -2.2 - 7.6 + 9.8 = -9.8 + 9.8 = 0

Equation (4) is satisfied exactly.

Since the system of 4 equations with 3 variables yields a unique consistent solution $(x = -2.2, y = 3.8, z = 1.4)$ , the number of solutions is 1.

Final answer

Common trap

Concluding there are 0 solutions simply because there are 4 equations in 3 variables (an overdetermined system), without checking whether the fourth equation is consistent.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2014 Computer Science and Information Technology (CS) Session 1
Question: 14 of 65, printed as CS Q.4
Section: Computer Science and Information Technology

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