Consider the following system of linear equations in variables x, y, and z, where a, b, an… — GRE General Test

Consider the following system of linear equations in variables

x

y

, and

z

, where

a

b

, and

c

are real constants:

\begin{aligned} 2x - y + 3z &= a \\ x + 2y - z &= b \\ 7x + 4y + 3z &= c \end{aligned}

Which of the following statements must be true? Select all such statements.

If $c = 2a + 3b$ , the system has infinitely many solutions.Answer
There exist no real values of $a$ , $b$ , and $c$ for which the system has a unique solution.Answer
C
If $a = 0$ , $b = 0$ , and $c = 0$ , the system has $(0, 0, 0)$ as its only solution.
D
If $c \neq 2a + 3b$ , the system has infinitely many solutions.
E
If $a = 1$ , $b = 2$ , and $c = 7$ , the system has at least one solution.

Answer

The correct statements are: 'If

c = 2a + 3b

, the system has infinitely many solutions' and 'There exist no real values of

a

b

, and

c

for which the system has a unique solution.'

The correct options accurately reflect the structural properties of the system. First, scaling the first equation by 2 and the second by 3 yields

7x + 4y + 3z = 2a + 3b

. Comparing this with the third equation,

7x + 4y + 3z = c

, shows that when

c = 2a + 3b

, the third equation provides no new constraints, leaving 2 independent equations in 3 variables and thus producing infinitely many solutions. Second, because the coefficient matrix has linearly dependent rows, its rank is 2 (less than the 3 variables), making a unique solution impossible regardless of the constants

a

b

, and

c

Step-by-Step Solution

Analyze the linear dependence of the left-hand sides of the equations.

Observe that

2(2x - y + 3z) + 3(x + 2y - z) = (4x + 3x) + (-2y + 6y) + (6z - 3z) = 7x + 4y + 3z

Finding a linear combination of the first two equations that produces the left-hand side of the third equation allows us to analyze system consistency.

Determine the condition for consistency.

The system is consistent if and only if

2a + 3b = c

c = 2a + 3b

, the third equation is a linear combination of the first two, resulting in a system of 2 independent equations in 3 variables, which yields infinitely many solutions.

Evaluate the possibility of a unique solution.

The rank of the coefficient matrix is 2, which is strictly less than the number of variables (3).

A system of linear equations has a unique solution if and only if the rank of the coefficient matrix equals the number of variables. Thus, no choice of

a, b, c

can produce a unique solution.

Verify specific numerical options.

For

a=0, b=0, c=0

c = 2(0)+3(0)=0

, giving infinitely many solutions. For

a=1, b=2, c=7

2(1)+3(2)=8 \neq 7

, giving zero solutions.

Testing specific constant values confirms consistency or inconsistency based on whether

c = 2a + 3b

is satisfied.

Key Concept

Consistency and Number of Solutions in 3x3 Systems of Linear Equations

Question

Answer

Step-by-Step Solution

Key Concept

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