General Form

Convert linear equations to general form Ax + By + C = 0. Math 10-C Alberta.

Lesson 6.5 of Linear Relations and Functions in Math 10C — Alberta curriculum lessons.

What you'll learn in this lesson

• What is General Form?. Learn the definition of general form, the three rules that make an equation valid in general form, and how it handles special cases like horizontal and vertical lines.
• Converting Between Forms. Practice converting slope-intercept and slope-point equations into general form, and converting general form back to slope-intercept form to read off slope and intercepts.
• Finding Intercepts from General Form. Discover how setting each variable to zero instantly gives you the x-intercept and y-intercept, then use those two points to graph the line.
• Application — The Ribbon Problem. Apply general form to a real-world cutting problem, generate a data table, find the slope-intercept equation from the table, and verify it matches the original constraint equation.

Inside the lesson — a free preview

The Definition

What the Variables Mean. In Ax + By + C = 0:

A is a whole number and must be greater than or equal to zero (A ≥ 0).

B and C are integers (positive, negative, or zero).

No fractions are allowed anywhere in the equation.

Standard Form vs. General Form. Some textbooks write Ax + By = C and call it standard form.

The only difference is whether the constant is on the left or the right side of the equals sign.

General form puts everything equal to zero: Ax + By + C = 0.

Rules for General Form

Rule 1 — No Fractions. If your equation has fractions, multiply every term by the LCD to clear them.

Example: y = -(2)/(3)x + 6 has the fraction -(2)/(3).

Multiply every term by 3: 3y = -2x + 18.

Rearrange: 2x + 3y - 18 = 0. No fractions remain.

Rule 2 — A Must Be Positive. A is the coefficient of x and it must satisfy A ≥ 0.

If clearing fractions or rearranging gives a negative A, multiply the entire equation by -1 to flip every sign.

Example: -3x + y + 4 = 0 has A = -3 (negative — invalid).

Multiply by -1: 3x - y - 4 = 0. Now A = 3 ≥ 0.

Rule 3 — All Terms on One Side, Equal to Zero. Move every term to the left side so the right side equals 0.

The equation must look like Ax + By + C = 0, not Ax + By = -C.

Quick check: Is A a non-negative whole number? Are B and C integers? Are there no fractions? Does the right side equal 0? If yes to all four, you have valid general form.

Worked example: Intercepts of 2x - 3y - 6 = 0

x-intercept — Set y = 0. 2x - 3(0) - 6 = 0

2x - 6 = 0

2x = 6

x = 3

x-intercept: (3, 0)

y-intercept — Set x = 0. 2(0) - 3y - 6 = 0

-3y - 6 = 0

-3y = 6

y = -2

y-intercept: (0, -2)

Graphing the Line. Plot the two intercepts (3, 0) and (0, -2), then draw the line through them.

Slope check: Converting 2x - 3y - 6 = 0 to slope-intercept form gives y = (2)/(3)x - 2.

The y-intercept is -2 and the slope is (2)/(3). From (0, -2) to (3, 0): rise = 2, run = 3. This matches the slope (2)/(3).

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