Standard Form & Converting Between Forms
What You’ll Learn
Section titled “What You’ll Learn”In this lesson you’ll learn standard form of a linear equation, why it’s useful, and how to convert between slope-intercept, point-slope, and standard forms.
The Concept
Section titled “The Concept”Standard form is Ax + By = C, where A, B, and C are integers, A ≥ 0, and A and B are not both zero. It’s useful for:
- Systems of equations (easier to solve with elimination)
- Finding intercepts quickly (x-intercept: C/A when y = 0; y-intercept: C/B when x = 0)
- Keeping coefficients integer with no fractions
Conversions:
- Slope-intercept y = mx + b → standard: mx − y = −b (multiply by −1 if needed for A > 0)
- Point-slope y − y₁ = m(x − x₁) → distribute m, move terms to one side, clear fractions if needed
- Standard → slope-intercept: solve for y (isolate y term)
Example conversions:
- y = 3x − 5 → 3x − y = 5
- y − 4 = −2(x + 1) → y − 4 = −2x − 2 → y + 2x = 2 → 2x + y = 2
- 4x − 5y = 20 → −5y = −4x + 20 → y = (4/5)x − 4 (slope-intercept)
Here’s 3x + 2y = 12 in standard form. Notice how the intercepts are easy to find:
Worked Example
Section titled “Worked Example”Convert y = −(3/2)x + 6 to standard form.
- y + (3/2)x = 6
- Multiply by 2 to clear fraction: 2y + 3x = 12
- 3x + 2y = 12 (A = 3 > 0)
Another: Convert point-slope y − 7 = 4(x − 2) to standard.
- y − 7 = 4x − 8
- y = 4x − 1
- −4x + y = −1 → 4x − y = 1
Real-World Application
Section titled “Real-World Application”Standard form is common in budgeting, business, or planning:
- Budget constraint: 3x + 5y = 150 (3 dollars food + 5 dollars gas = 150 dollars total budget)
- Two plans: 4x + 2y = 100 and 5x + 3y = 120 (solve system for break-even)
- Linear programming (intro): constraints in standard form help find max/min
Converting forms lets you pick the best view: slope-intercept for rate/start, standard for intercepts/systems.