Question

Consider the following system of equations in three variables \(x, y, z\): \[ -x - y - z = 3 \] \[ x + y + z = 10 \] \[ 2x - 3y = 6 \] This system of equations has

• A. no combination of values of \((x, y, z)\) that satisfy this system simultaneously
• B. only one combination of values of \((x, y, z)\) that satisfy this system simultaneously
• C. only two combinations of values of \((x, y, z)\) that satisfy this system simultaneously
• D. infinitely many combinations of values of \((x, y, z)\) that satisfy this system simultaneously

Hint:

If you encounter a contradiction while solving a system of equations, it indicates that the system has no solution.

Answer 1

The Correct Option is A

Step 1: Solve the system of equations.
We begin by adding the first and second equations: \[ (-x - y - z) + (x + y + z) = 3 + 10 \] This simplifies to: \[ 0 = 13 \] This is a contradiction, which means the system has no solution.
Step 2: Conclusion.
Since we reached a contradiction in the first step, we can conclude that no combination of values of \((x, y, z)\) satisfies the system simultaneously.
Hence, the correct answer is (A).