The solution of the system of the given linear equations is: 2x + 3y − z = 5, x − 2y + 4z = −2, 3x + y + 2z =

Question

The solution of the system of the given linear equations is: 2x + 3y − z = 5, x − 2y + 4z = −2, 3x + y + 2z = 7

Examoogle · Accepted Answer

The correct answer is Option C: $x = \frac{44}{7}, y = \frac{-27}{7}, z = -4$. Check the given options by substituting the values into the equations. For instance, in Eq 2 (x - 2y + 4z = -2), substituting Option C: (44/7) - 2(-27/7) + 4(-4) = (44/7) + (54/7) - 16 = (98/7) - 16 = 14 - 16 = -2. It matches.

Step-by-Step Solution & Concepts:
Shortcut Trick: Check the given options by substituting the values into the equations. For instance, in Eq 2 (x - 2y + 4z = -2), substituting Option C: (44/7) - 2(-27/7) + 4(-4) = (44/7) + (54/7) - 16 = (98/7) - 16 = 14 - 16 = -2. It matches.

Formula Used: $$D = \begin{vmatrix} 2 & 3 & -1 \ 1 & -2 & 4 \ 3 & 1 & 2 \end{vmatrix}, x = \frac{D_x}{D}, y = \frac{D_y}{D}, z = \frac{D_z}{D}$$

Given: System of linear equations: 2x + 3y - z = 5, x - 2y + 4z = -2, 3x + y + 2z = 7

Steps: Calculate determinant D: Expanding the determinant of the coefficients: $$D = 2(-4 - 4) - 3(2 - 12) - 1(1 + 6) = 2(-8) - 3(-10) - 1(7) = -16 + 30 - 7 = 7$$.. Calculate determinant Dx: Replacing the x-column with constants (5, -2, 7): $$D_x = \begin{vmatrix} 5 & 3 & -1 \ -2 & -2 & 4 \ 7 & 1 & 2 \end{vmatrix} = 5(-4-4) - 3(-4-28) - 1(-2+14) = 5(-8) - 3(-32) - 1(12) = -40 + 96 - 12 = 44$$.. Calculate determinant Dy: Replacing the y-column with constants: $$D_y = \begin{vmatrix} 2 & 5 & -1 \ 1 & -2 & 4 \ 3 & 7 & 2 \end{vmatrix} = 2(-4-28) - 5(2-12) - 1(7+6) = 2(-32) - 5(-10) - 1(13) = -64 + 50 - 13 = -27$$.. Calculate z: Using the equation $3x + y + 2z = 7$: $$3(\frac{44}{7}) + (\frac{-27}{7}) + 2z = 7 \implies \frac{132-27}{7} + 2z = 7 \implies \frac{105}{7} + 2z = 7 \implies 15 + 2z = 7 \implies 2z = -8 \implies z = -4$$.

Answer: C is correct because the system of linear equations yields x = 44/7, y = -27/7, and z = -4.

Exam Tip: Solving systems of equations is fundamental in Matrix Algebra and Linear Programming, which often appear in technical competitive exams.

Examoogle · Answer

Option A: $x = \frac{-44}{7}, y = \frac{27}{7}, z = -4$

Examoogle · Answer

Option B: $x = \frac{-44}{7}, y = \frac{-27}{7}, z = 4$

Examoogle · Answer

Option D: $x = \frac{44}{7}, y = \frac{-27}{7}, z = -8$

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