Which of the following is the equation of the line through \((3,\,-2)\) that is parallel to the line \(2x+y=7\)?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(2x+y=4\)
Parallel lines have equal slopes. Lines \(2x+y=c\) all have the same slope as \(2x+y=7\) (only the constant changes), so the required line is \(2x+y=c\) and the given point fixes \(c\).
Slope of \(2x+y=7\): \(y=-2x+7\), so the slope is \(-2\)
Required line: \(2x+y=c\)
It passes through \((3,\,-2)\): \(c=2(3)+(-2)=4\)
Equation: \(2x+y=4\)
Why the other options are incorrect
Option (1) \(x-2y=7\) — Incorrect. It passes through \((3,\,-2)\), but its slope is \(\frac{1}{2}\), not \(-2\).
Option (3) \(2x+y=-4\) — Incorrect. It is parallel, but at \((3,\,-2)\) the left side is \(4\), not \(-4\).
Option (4) \(x+2y=-1\) — Incorrect. It passes through \((3,\,-2)\), but its slope is \(-\frac{1}{2}\), not \(-2\).