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

Consider the following system of linear equations in variables

x

y

, and

z

, where

a

is a real constant:

\begin{aligned} x + y + z &= 6 \\ x + 2y + 3z &= 10 \\ x + 2y + (a^2 - 1)z &= a + 8 \end{aligned}

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

If $a = 2$ , the system has infinitely many solutions.Cevap
If $a = -2$ , the system has no solution.Cevap
If $a = 2$ , every solution to the system satisfies $2x + y = 8$ .Cevap
D
If $a = 3$ , the unique solution has $y = 3$ .
E
The system has a unique solution for all real values of $a$ except $a = 2$ .

Cevap

The correct statements are that

a = 2

yields infinitely many solutions,

a = -2

results in no solution, and for

a = 2

every solution satisfies

2x + y = 8

The reduced equation

(a^2 - 4)z = a - 2

determines the behavior of the system. Setting

a = 2

gives

0 = 0

, leading to infinitely many solutions where

x = z + 2

and

y = 4 - 2z

, which identically satisfies

2x + y = 8

. Setting

a = -2

gives

0 = -4

, an inconsistency yielding no solutions.

Adım Adım Çözüm

Eliminate

x

and

y

using elimination between the second and third equations.

(x + 2y + (a^2 - 1)z) - (x + 2y + 3z) = (a + 8) - 10 \implies (a^2 - 4)z = a - 2

Isolating the parameter dependence onto a single variable

z

reveals existence and uniqueness conditions.

Analyze the equation

(a - 2)(a + 2)z = a - 2

for key parameter values.

a = 2

0 \cdot z = 0

(infinitely many solutions). If

a = -2

0 \cdot z = -4

(no solution). If

a \neq \pm 2

z = \frac{1}{a + 2}

(unique solution).

Determining system consistency depends on whether the leading coefficient and right-hand side evaluate to zero.

Express

x

and

y

in terms of

z

for the consistent case

a = 2

Subtracting the first equation from the second gives

y + 2z = 4 \implies y = 4 - 2z

. Substituting into the first gives

x = z + 2

Parameterizing the solution set allows verification of linear combinations.

Evaluate the linear combination

2x + y

when

a = 2

2x + y = 2(z + 2) + (4 - 2z) = 2z + 4 + 4 - 2z = 8

Verifies that

2x + y = 8

is an invariant across all parametric solutions.

Anahtar Kavram

Parametric Analysis of 3x3 Systems of Linear Equations

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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