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

Consider the following system of three linear equations in variables

x

y

, and

z

, where

k

is a real constant:

\begin{aligned} 2x + 3y - z &= 11 \\ x - 2y + 4z &= -3 \\ 4x - y + 7z &= k \end{aligned}

For what value of

k

does the system have at least one solution

(x, y, z)

A
$-5$
$5$ Answer
C
$8$
D
$17$
E
$23$

Answer

5

The left-hand side of the third equation is a linear combination of the first two equations:

1 \cdot (2x + 3y - z) + 2 \cdot (x - 2y + 4z) = 4x - y + 7z

. For the linear system to be consistent and possess at least one solution, the same linear combination must hold for the constant terms on the right-hand side:

1(11) + 2(-3) = 11 - 6 = 5

. Therefore, the value of

k

must be

5

Step-by-Step Solution

Analyze the variable coefficients across the three equations for linear dependence.

Notice that the coefficients of the third equation can be expressed as a linear combination of the first two equations.

If the left-hand side of the third equation is a linear combination of the first two equations, the system will only be consistent if the right-hand side constants satisfy the exact same linear combination.

Determine the multiplier needed to produce the third equation's left-hand side.

Multiply the second equation by

2

and add it to the first equation:

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

This yields the exact expression

4x - y + 7z

present on the left-hand side of the third equation.

Apply the identical combination to the right-hand side constant terms.

The combined constant value is

11 + 2(-3) = 11 - 6 = 5

For the system to have at least one solution (i.e., to avoid contradiction and be consistent),

k

must equal this computed value of

5

Key Concept

Linear Dependence and Consistency in 3x3 Linear Systems

Estimated Time:2m 0s

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