Which line has slope −1 and passes through (2,3)?

Correct — A. Use the point-slope form y − y₁ = m(x − x₁).

Put in m = −1 and (x₁, y₁) = (2, 3): y − 3 = −1(x − 2)
Expand the right side: y − 3 = −x + 2
Add 3 to both sides: y = −x + 5

Check: at x = 2, y = −2 + 5 = 3, so (2, 3) lies on it → option (a)

Why the others are wrong

• (b)
  y = x + 5 — Slope is +1, not −1: the coefficient of x is positive. At x = 2 it gives y = 7, so it also misses (2, 3).
• (c)
  y = 5x + 2 — Slope is 5, the coefficient of x, not −1. At x = 2 it gives y = 12, not 3, so the line misses the point as well.
• (d)
  y = −5x + 2 — Slope is −5, not −1. At x = 2 it gives y = −10 + 2 = −8, so the line does not pass through (2, 3).

In y = mx + c, m is the slope and c is the y-intercept, where the line cuts the y-axis.

A slope fixes the line's tilt but not its position. One point fixes the position: substitute it to find c. Here 3 = −1 × 2 + c gives c = 5.

Key facts

• Point-slope form: y − y₁ = m(x − x₁).
• Slope through (x₁, y₁) and (x₂, y₂) = (y₂ − y₁) ⁄ (x₂ − x₁).
• Perpendicular lines have slopes whose product is −1.
• Parallel lines have equal slopes.

Common traps

• Sign slip while expanding: −1(x − 2) is −x + 2, not −x − 2, which would give y = −x + 1.
• Testing only the point: y = x + 1 also passes through (2, 3) but has slope +1, so both conditions must hold.

No directly related past PYQ was found.