Point-slope and two-point forms

The equation of a line is a condition on (x, y) that is true for every point on the line and false for every point off it.

A horizontal line at distance a from the x-axis is y = a or y = −a; a vertical line at distance b from the y-axis is x = b or x = −b. Through (−2, 3) the horizontal line is y = 3 and the vertical line is x = −2.

Point-slope form: if a line has slope m and passes through (x₀, y₀), its equation is y − y₀ = m(x − x₀). With slope −4 through (−2, 3) this reads y − 3 = −4(x + 2); expanding, y − 3 = −4x − 8, so 4x + y + 5 = 0.

Two-point form: the line through (x₁, y₁) and (x₂, y₂) is y − y₁ = ((y₂ − y₁)/(x₂ − x₁))(x − x₁), which says the slope from P₁ to any point P equals the slope P₁P₂.

Through (1, −1) and (3, 5): the slope is 6/2 = 3, so y + 1 = 3(x − 1), or −3x + y + 4 = 0, which is y = 3x − 4.

Lines are concurrent when they pass through one common point. The lines 2x + y − 3 = 0 and 3x − y − 2 = 0 meet at (1, 1); the line 5x + ky − 3 = 0 passes through (1, 1) when 5 + k − 3 = 0, that is k = −2.