Consider the following system of linear equations in variables x and y, where k is a real … — GRE General Test

Consider the following system of linear equations in variables

x

and

y

, where

k

is a real constant:

\begin{aligned} kx + 4y &= 8 \\ x + ky &= k + 2 \end{aligned}

Which of the following statements are true? Select all that apply.

If $k = 2$ , the system has infinitely many solutions.Answer
If $k = -2$ , the system has no solutions.Answer
If $k = 0$ , the unique solution to the system is $(x, y) = (2, 2)$ .Answer
D
If $k = 3$ , the system has no solutions.
E
If $k = 1$ , the solution to the system satisfies $x + y = 4$ .

Answer

The correct statements are that setting

k = 2

results in infinitely many solutions, setting

k = -2

results in no solutions, and setting

k = 0

yields the unique solution

(2, 2)

The system has a coefficient matrix determinant of

k^2 - 4

. Setting

k = 2

produces identical equations (

x + 2y = 4

), giving infinitely many solutions. Setting

k = -2

produces parallel equations with different constants (

-x + 2y = 4

-x + 2y = 0

), giving no solutions. Setting

k = 0

reduces the system directly to

y = 2

and

x = 2

, confirming the unique point

(2, 2)

Step-by-Step Solution

Analyze the determinant of the coefficient matrix to identify conditions for unique vs. non-unique solutions.

The coefficient matrix determinant is

\Delta = k\cdot k - 4\cdot 1 = k^2 - 4 = (k - 2)(k + 2)

\Delta \neq 0

(i.e.,

k \neq \pm 2

), the system has a unique solution. If

\Delta = 0

(i.e.,

k = 2

k = -2

), the lines are either identical or parallel.

Test

k = 2

in the system.

Equation 1 becomes

2x + 4y = 8 \implies x + 2y = 4

. Equation 2 becomes

x + 2y = 4

Since the equations are identical, the lines coincide, resulting in infinitely many solutions.

Test

k = -2

in the system.

Equation 1 becomes

-2x + 4y = 8 \implies -x + 2y = 4

. Equation 2 becomes

x - 2y = 0 \implies -x + 2y = 0

The slopes are equal (

1/2

) but the y-intercepts differ (

2

0

), meaning the lines are parallel and distinct, yielding zero solutions.

Test

k = 0

and

k = 1

to evaluate remaining choices.

For

k = 0

4y = 8 \implies y = 2

and

x + 0 = 2 \implies x = 2

, giving

(2, 2)

. For

k = 1

, Equation 2 is directly

x + y = 3

This verifies that the statement for

k = 0

is correct, while the statement for

k = 1

claiming

x + y = 4

is false.

Key Concept

Parametric Linear Systems and Conditions for Solvability

Question

Answer

Step-by-Step Solution

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