Key Concepts and Formulas

• System of Linear Equations: A set of linear equations with the same variables. The solution to the system is the set of values for the variables that satisfy all equations simultaneously.
• Elimination Method: A method for solving systems of linear equations by adding or subtracting multiples of the equations to eliminate one or more variables.
• Equation of a Straight Line: The general form of a straight line is $Ax + By + C = 0$, where $A$, $B$, and $C$ are constants.

Step-by-Step Solution

Step 1: State the given system of equations. We are given the following system of linear equations: $x - 2y + kz = 1 \quad \text{...(1)}$ $2x + y + z = 2 \quad \text{...(2)}$ $3x - y - kz = 3 \quad \text{...(3)}$ Our goal is to find the relationship between $x$ and $y$ by eliminating $z$ and $k$.

Step 2: Eliminate $kz$ by adding equations (1) and (3). Observe that equation (1) has a $+kz$ term and equation (3) has a $-kz$ term. Adding these equations will eliminate both $k$ and $z$ from these terms simultaneously. Adding equation (1) and equation (3): $(x - 2y + kz) + (3x - y - kz) = 1 + 3$

Step 3: Simplify the resulting equation. Combine like terms: $(x + 3x) + (-2y - y) + (kz - kz) = 4$ $4x - 3y = 4$

Step 4: Rewrite the equation in the standard form of a linear equation. Subtract 4 from both sides to get the equation in the form $Ax + By + C = 0$: $4x - 3y - 4 = 0$ This is the equation of the straight line on which the point $(x, y)$ lies.

Common Mistakes & Tips

• Strategic Elimination: Look for opportunities to eliminate variables directly by adding or subtracting equations. In this case, the $+kz$ and $-kz$ terms made elimination straightforward.
• Avoid Unnecessary Calculations: Do not attempt to solve for $x, y, z,$ and $k$ individually unless required. This problem only asks for a relationship between $x$ and $y$, so finding a direct relationship is more efficient.
• The Condition $z \ne 0$: The condition $z \ne 0$ is given to ensure that a solution exists with $z$ not equal to zero. However, this condition does not affect the elimination process to find the relationship between $x$ and $y$.

Summary

By adding equations (1) and (3), we eliminated $z$ and $k$ simultaneously, resulting in the equation $4x - 3y = 4$, which can be rewritten as $4x - 3y - 4 = 0$. This is the equation of the straight line on which the point $(x, y)$ must lie.

Final Answer

The final answer is \boxed{4x – 3y – 4 = 0}, which corresponds to option (A).