L8 · Three Forms of a Line: Slope-Intercept, General, Slope-Point

Learning goal

“I can rewrite a linear relation among slope-intercept form, general form (integers, A >= 0), and slope-point form; recognize equivalent equations of the same line; and extract slope and intercepts from any form.”

Slope-intercept form — y = mx + b: displays the slope m and the y-intercept b directly.

General form — Ax + By + C = 0, with A, B, C integers and A >= 0 by convention (if A = 0, it is written By + C = 0 with B > 0).

Slope-point form — y - y1 = m(x - x1): displays the slope m and one point (x1, y1) on the line.

Equivalent linear relations — different equations that have exactly the same graph (every point satisfying one satisfies the others).

Worked example

Convert y = 2x + 3 to general form. Move everything to one side: subtract 2x and 3 - or, keeping A positive, write 2x - y + 3 = 0. Check the convention: A = 2, B = -1, C = 3 - all integers, A >= 0. Verify equivalence with a point: (0, 3) is on y = 2x + 3; in the general form: 2(0) - 3 + 3 = 0. With a fraction: y = (1/2)x - 4. First clear the fraction by multiplying everything by 2: 2y = x - 8. Then rearrange: x - 2y - 8 = 0 (A = 1 >= 0).

Check yourself — explain it

Multiplying every term by 2 when clearing a fraction does not change the line because...

Paired problem — example still visible

Convert y = -3x + 1 to general form. -> 3x + y - 1 = 0 (add 3x to both sides: 3x + y = 1, then subtract 1)

Faded completion — last step is yours

"Convert y = 4x - 5 to general form: ___ x - y - ___ = 0." -> 4x - y - 5 = 0

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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Grade 10 — Mathematics 10C › C10U07 Linear Relations & Functions

L8 · Three Forms of a Line: Slope-Intercept, General, Slope-Point