Straight lines and coordinate lengths

Gradient and intercept determine a non-vertical line

Through (1,5) and (4,11), the gradient is (11-5)/(4-1) = 2. Substitute one point in y = 2x + c: 5 = 2 + c, so c = 3. The line is y = 2x + 3. Check the other point before accepting it.

A horizontal line through (1,5) has equation y = 5. A vertical line through that point has equation x = 1 and cannot be expressed as y = mx + c with finite m. Do not divide by its zero horizontal difference.

The distance comes from a right triangle

Between A(−2,1) and B(4,9), the horizontal change is 6 and the vertical change 8. Pythagoras gives

Reversing both point orders changes the signs of the differences but not their squares. Adding the changes to get 14 would measure a horizontal-then-vertical route, not the straight segment.

Use coordinates to justify geometry

A(0,0), B(6,0), C(6,4), D(0,4) form a rectangle: AB and CD are horizontal, BC and AD vertical, and the adjacent directions are perpendicular. Its diagonals both have length square root of 52. Equal diagonals alone would not establish every rectangle property; the coordinate directions supply the right angles here.

Check your understanding 1 Substitute in the original data

Find the line through (−1, 7) and (3, −1), and the distance between the points.

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The gradient is −8/4 = −2. Substitution gives c = 5, so y = −2x + 5. The distance is square root of (4² + (-8)²) = square root of 80 = 4 square root of 5. Check both given points in the line equation.

A negative gradient does not measure length

The line’s gradient is −2, while its segment length is positive. A gradient describes change per horizontal unit; a length measures distance. The two quantities have different meanings and need not share a sign or unit.