Which of the following points lies on the same side as the origin, with…

Concept

A line ax + by + c = 0 divides the plane into two half-planes. Define the signed value f(x, y) = ax + by + c for any point. Two points lie on the SAME side of the line exactly when f(x, y) has the SAME sign at both points; the signs differ when the points lie on OPPOSITE sides (f = 0 places a point ON the line).

Application

Write the line 3x + 7y = 2 as f(x, y) = 3x + 7y − 2, then evaluate it at the origin and at each candidate point — whichever point shares the origin's sign lies on the origin's side.

1. Origin: f(0, 0) = 3(0) + 7(0) − 2 = −2, which is negative, so the target side is the f < 0 side.

2. (3, 0): f = 3(3) + 7(0) − 2 = 9 − 2 = 7, positive — opposite side from the origin.

3. (1, 0): f = 3(1) + 7(0) − 2 = 3 − 2 = 1, positive — opposite side from the origin.

4. (1/2, 1/2): f evaluates to a positive result (3) at this point — opposite side from the origin.

5. (1/2, 0): f evaluates to a negative result at this point — the same sign as the origin, so this point lies on the same side.

Cross-check

Confirm independently using the inequality form directly: the origin's half-plane is where 3x + 7y < 2 (since 0 < 2 at the origin). At (1/2, 0), the sum 3x + 7y comes out below 2, satisfying this inequality directly — the same conclusion reached without shifting to the f(x, y) form.