Question

If $(x,y,z)$ is the solution of the equations $4x + y = 7$, $3y + 4z = 5$, $5x + 3z = 2$, then the value of $x + y + z$ equals

• A. $8$
• B. $6$
• C. $3$
• D. $0$
• E. $1$

Hint:

Reduce the system to two variables first, then solve step-by-step to avoid confusion.

Answer 1

Answer: (E)

Concept: We solve a system of three linear equations using substitution/elimination and then compute the required sum.

Step 1: Write the system clearly. \[ (1)\; 4x + y = 7 \] \[ (2)\; 3y + 4z = 5 \] \[ (3)\; 5x + 3z = 2 \]

Step 2: Express one variable from equation (1). \[ y = 7 - 4x \]

Step 3: Substitute into equation (2). \[ 3(7 - 4x) + 4z = 5 \] \[ 21 - 12x + 4z = 5 \] \[ -12x + 4z = -16 \] Divide by 4: \[ -3x + z = -4 \cdots (4) \]

Step 4: Use equation (3). \[ 5x + 3z = 2 \cdots (5) \]

Step 5: Solve equations (4) and (5). From (4): \[ z = -4 + 3x \] Substitute into (5): \[ 5x + 3(-4 + 3x) = 2 \] \[ 5x - 12 + 9x = 2 \] \[ 14x = 14 \] \[ x = 1 \]

Step 6: Find $z$. \[ z = -4 + 3(1) = -1 \]

Step 7: Find $y$. \[ y = 7 - 4(1) = 3 \]

Step 8: Compute required sum. \[ x + y + z = 1 + 3 - 1 = 3 \]
Final Answer: \[ \boxed{3} \]