Solve system: y = 2x + 1 and y = −x + 4

Correct — A. Both equations give y, so equate the right-hand sides.

2x + 1 = −x + 4
Add x to both sides: 3x + 1 = 4
Subtract 1: 3x = 3
Divide by 3: x = 1

Substitute into y = 2x + 1: y = 2 + 1 = 3
Check in y = −x + 4: −1 + 4 = 3 ✓

The lines meet at (1, 3) → option (a)

Why the others are wrong

• (b)
  (2, 5) — (2, 5) fits y = 2x + 1, since 2 × 2 + 1 = 5, but fails y = −x + 4: −2 + 4 = 2, not 5. It lies on the first line only.
• (c)
  (3, 7) — (3, 7) fits y = 2x + 1 but not y = −x + 4, which gives −3 + 4 = 1. Moving −x across without changing its sign gives x = 3 and this point.
• (d)
  (0, 1) — (0, 1) is where y = 2x + 1 crosses the y-axis. In y = −x + 4, x = 0 gives y = 4, so this point is not on the second line.

A system's solution is the point that satisfies both equations at once — where the two lines cross.

The slopes here are 2 and −1. Different slopes mean the lines are not parallel, so they meet at exactly one point.

When both equations are already written as y = …, setting them equal removes y in one step.

Key facts

• A solution of a system must satisfy every equation in it.
• Lines y = m₁x + c₁ and y = m₂x + c₂ with m₁ ≠ m₂ meet at exactly one point.
• Equal slopes with different intercepts give parallel lines and no solution.
• Equal slopes and equal intercepts give the same line and infinitely many solutions.

Common traps

• Testing an option in one equation only: every option here satisfies y = 2x + 1.
• Moving −x to the left as −x instead of +x, which gives x = 3 and the point (3, 7).

No directly related past PYQ was found.