直線方程：斜率與三式

§Text Passage

1Summary 重點摘要

Slope, point-slope and general form: two facts determine a straight line.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: The slope formula and the four tilt typesThe slope is m = (y₂-y₁)/(x₂-x₁): the vertical 'rise' over the horizontal 'run'. For A(1, 2) and B(3, 8): m = (8-2)/(3-1) = 3, meaning the line climbs 3 units for every 1 unit across. Classify by sign: m > 0 rises to the right, m < 0 falls, m = 0 is horizontal, and a zero denominator gives a vertical line with undefined slope. The point chosen must lead in both the numerator and the denominator; a mismatch adds a minus sign and describes a line leaning the other way.
2. Step 2: Point-slope form — one point plus a slopeA line through (x₁, y₁) with slope m has equation y - y₁ = m(x - x₁), because every point on the line forms the same slope m with the fixed point. Through (2, 1) with slope 3: y - 1 = 3(x - 2), which simplifies to y = 3x - 5. Substitute x = 2 before submitting and confirm y = 1 — one self-check catches sign slips at once. The fixed point belongs in (x₁, y₁); putting numbers in place of the variables x and y turns the equation into a numeric statement.
3. Step 3: Two points — slope first, then point-slopeGiven two points, compute the slope first, then apply the point-slope form with either point. Through A(1, 2) and B(3, 8): m = 3, then y - 2 = 3(x - 1) using A, simplifying to y = 3x - 1; checking with B gives 3(3) - 1 = 8, correct. Two points fix one line, so either choice gives the same equation — the points already share it. Starting from the point-slope form before computing the slope, or slipping a sign while simplifying, are the usual errors; verifying the other point exposes them at almost no cost.
4. Step 4: General form and two quick readingsFrom Ax + By + C = 0, read the slope as -A/B and the y-intercept as -C/B (when B is not 0). For 2x + 3y - 6 = 0: slope -23, y-intercept 2. The sketching shortcut is to find both intercepts: set y = 0 to get the x-intercept 3, set x = 0 to get the y-intercept 2, then join (3, 0) and (0, 2). Both readings carry a minus sign that is easy to drop. A vertical line x = a has B = 0, so neither reading exists — it is drawn as a straight line through (a, 0).
5. Step 5: Parallel and perpendicular — two slope rulesParallel lines share equal slopes: through (0, 3) parallel to y = 2x + 1 is y = 2x + 3. Perpendicular lines multiply to -1: the line perpendicular to y = 2x + 1 has slope -12 — flip the fraction and change the sign. The exam routine is fixed: read the slope of the given line, apply the parallel or perpendicular rule to get the new slope, then write the equation in point-slope form. Flipping without changing the sign is the classic slip; a horizontal and a vertical line are perpendicular too, but the product rule does not apply to that pair.
6. Step 6: Intersections by simultaneous equationsThe intersection point satisfies both equations, so solve them simultaneously. For y = 3x - 1 and y = -x + 7: set 3x - 1 = -x + 7, so 4x = 8, x = 2, and substituting back gives y = 5 — the point (2, 5). The same method meets a line with a curve: substitute the line into the curve and solve the resulting quadratic. Always substitute the answer back into the other equation to confirm both are satisfied. Sign errors while rearranging are common; when a question asks for the number of intersections, use the discriminant instead of hunting for the points.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Slope is rise over run.
• Point-slope is the daily tool.
• General form: slope=-A/B, y-intercept=-C/B.
• Parallel: equal slopes; perpendicular: negative reciprocal.
• Intersections by simultaneous equations.