If the simultaneous equations 2x − 5y = 5 and 2x − y = 9 both hold, what is the value of the expression x − y?

Introduction / Context:This problem involves solving a system of two linear equations in two variables and then using the solution to compute a simple expression, x − y. Such questions test your ability to manipulate simultaneous equations, often using elimination or substitution methods, and then interpret the solution in the form requested by the problem.

Given Data / Assumptions:

• Equations: 2x − 5y = 5 and 2x − y = 9.
• x and y are real numbers.
• We are asked for x − y, not for x and y separately.
• All arithmetic is exact.

Concept / Approach:The two equations have the same coefficient for x, which makes elimination particularly convenient. By subtracting one equation from the other, we can remove x and solve directly for y. Once y is found, we substitute back into either equation to find x. With x and y known, calculating x − y is straightforward. This method avoids unnecessary complexity and highlights the usefulness of elimination in solving linear systems.

Step-by-Step Solution:Write the two equations together: 2x − 5y = 5 and 2x − y = 9.Subtract the second equation from the first to eliminate x: (2x − 5y) − (2x − y) = 5 − 9.The left side simplifies to 2x − 5y − 2x + y = −4y.The right side is 5 − 9 = −4, so −4y = −4.Divide both sides by −4 to find y: y = 1.Substitute y = 1 into 2x − y = 9: 2x − 1 = 9, so 2x = 10 and x = 5.Finally compute x − y: x − y = 5 − 1 = 4.

Verification / Alternative check:Check the solution (x, y) = (5, 1) in both original equations. For 2x − 5y: 2 * 5 − 5 * 1 = 10 − 5 = 5, which matches the first equation's right-hand side. For 2x − y: 2 * 5 − 1 = 10 − 1 = 9, which matches the second equation's right-hand side. Since both equations are satisfied, the solution is correct and x − y = 4 is confirmed.

Why Other Options Are Wrong:

• Values 2, 3, 6, and 1 would require different combinations of x and y that do not satisfy both equations simultaneously.
• For example, if x − y = 2, you might have (x, y) = (3, 1), which does not satisfy either equation.
• Only x − y = 4 arises from the unique solution pair (5, 1) of the system.

Common Pitfalls:

• Making sign errors when subtracting one equation from the other.
• Substituting into the wrong equation or miscomputing 2x − 1 = 9.
• Stopping after finding x and y without correctly computing the requested expression x − y.

Final Answer:4