3.8Standard Form of Linear Equations
Ax + By = C — standard form organizes a linear equation with both variables on one side. It is perfect for finding intercepts quickly and for setting up systems of equations.
Why This Matters
Standard form Ax + By = C makes finding intercepts fast and is the preferred form in many real-world applications, including systems of equations and linear programming.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
What is standard form Ax + By = C, and how do you convert between standard form and slope-intercept form?
Lesson Overview
Standard form of a linear equation is Ax + By = C, where A, B, and C are integers, A is greater than or equal to 0, and there are no fractions. Standard form is useful because you can find both intercepts quickly: set y = 0 to find the x-intercept, and set x = 0 to find the y-intercept. To convert from slope-intercept to standard form, move the x-term to the left side and clear any fractions by multiplying through. To convert from standard form to slope-intercept, solve for y by isolating it on the left side.
Anatomy of Ax + By = C
Ax + By = C
A, B, C are integers; A ≥ 0; no fractions
Finding Intercepts
x-intercept: set y = 0, solve for x
y-intercept: set x = 0, solve for y
Converting to y = mx + b
Subtract Ax from both sides, then divide everything by B.
m = −A/B, b = C/B
Worked Examples
Find the x-intercept and y-intercept of 3x + 4y = 12, then graph the line.
Find the x-intercept: set y = 0.
3x + 4(0) = 12 → 3x = 12 → x = 4. x-intercept: (4, 0).
Find the y-intercept: set x = 0.
3(0) + 4y = 12 → 4y = 12 → y = 3. y-intercept: (0, 3).
Plot (4, 0) and (0, 3) on the coordinate plane.
Draw a straight line through both points.
Check with a third point: x = 2 → 3(2) + 4y = 12 → y = 3/2 = 1.5. Point (2, 1.5) should be on the line.
Example 1 — 3x + 4y = 12
Convert y = 2x − 5 to standard form.
Goal: get both variables on the left side with integer coefficients.
Start: y = 2x − 5
Subtract 2x from both sides: −2x + y = −5
Multiply every term by −1 to make A positive: 2x − y = 5
Check: A = 2, B = −1, C = 5. A is positive, all integers, no fractions. ✓
Verify: substitute (3, 1) → 2(3) − 1 = 5 ✓.
Convert y = (3/4)x + 2 to standard form.
Start: y = (3/4)x + 2
Multiply every term by 4 to clear the fraction: 4y = 3x + 8
Subtract 3x from both sides: −3x + 4y = 8
Multiply by −1 to make A positive: 3x − 4y = −8
Check: A = 3, B = −4, C = −8. All integers, A positive, no fractions. ✓
Convert 5x − 2y = 10 to slope-intercept form.
Goal: solve for y.
5x − 2y = 10
Subtract 5x from both sides: −2y = −5x + 10
Divide every term by −2: y = (5/2)x − 5
Identify: m = 5/2, b = −5.
Check: substitute x = 2 into original: 5(2) − 2y = 10 → y = 0. Check in slope-intercept: y = (5/2)(2) − 5 = 0 ✓.
Example 4 — 5x − 2y = 10
Write the standard form equation of the line through (2, 1) and (4, 5).
Step 1: Find the slope.
m = (5 − 1) / (4 − 2) = 4 / 2 = 2
Step 2: Write slope-intercept form using point (2, 1).
1 = 2(2) + b → b = 1 − 4 = −3. So y = 2x − 3.
Step 3: Convert to standard form.
Subtract 2x: −2x + y = −3
Multiply by −1: 2x − y = 3
Check with (4, 5): 2(4) − 5 = 3 ✓.
Guided Practice
Guided Practice Video: Standard Form of Linear Equations
Watch the guided practice walkthrough for standard form of linear equations, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Answers are in the Answer Key section.
Find the x-intercept and y-intercept of 2x + 5y = 10.
Hint: x-intercept: set y=0 → 2x=10 → x=5, so (5,0). y-intercept: set x=0 → 5y=10 → y=2, so (0,2).
Convert y = 3x − 7 to standard form.
Hint: Subtract 3x: −3x + y = −7. Multiply by −1: 3x − y = 7.
Convert y = (2/3)x − 4 to standard form.
Hint: Multiply by 3: 3y = 2x − 12. Subtract 2x: −2x + 3y = −12. Multiply by −1: 2x − 3y = 12.
Convert 4x + 2y = 8 to slope-intercept form.
Hint: Subtract 4x: 2y = −4x + 8. Divide by 2: y = −2x + 4. m = −2, b = 4.
Write the standard form equation of the line through (1, 3) and (3, 7).
Hint: m = (7−3)/(3−1) = 2. Using (1,3): 3 = 2(1)+b → b=1. y=2x+1. Standard form: 2x−y=−1.
Key Vocabulary
Standard Form
A linear equation written as Ax + By = C, where A, B, C are integers, A is greater than or equal to 0, and there are no fractions.
Example: 3x + 4y = 12 is in standard form. A=3, B=4, C=12.
x-intercept
The point where the line crosses the x-axis. Found by setting y = 0 in the equation.
Example: For 3x + 4y = 12: set y=0 → x=4. x-intercept is (4,0).
y-intercept
The point where the line crosses the y-axis. Found by setting x = 0 in the equation.
Example: For 3x + 4y = 12: set x=0 → y=3. y-intercept is (0,3).
Integer Coefficients
Coefficients that are whole numbers (no fractions or decimals). Standard form requires integer coefficients.
Example: 3x − 4y = −8 has integer coefficients. (3/4)x + y = 2 does not.
Converting Forms
Rewriting a linear equation from one form to another while keeping the same line. Standard form and slope-intercept form describe the same line.
Example: 3x − y = 5 converts to y = 3x − 5 (same line, different form).
Interactive Practice — 5 Questions
What is the x-intercept of 5x + 2y = 10?
Which is the standard form of y = −3x + 6?
Convert 6x − 3y = 9 to slope-intercept form.
Which equation is NOT in standard form?
A line has x-intercept (3, 0) and y-intercept (0, −6). What is the standard form equation?
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Find the x-intercept and y-intercept of 4x − 3y = 12. Then graph the line.
Convert y = −2x + 8 to standard form.
Convert y = (1/2)x − 3 to standard form (clear fractions first).
Convert 3x + 6y = 18 to slope-intercept form. Identify slope and y-intercept.
Challenge: Write the standard form equation of the line through (−1, 4) and (3, −4). Then find both intercepts and verify they satisfy the equation.
ChallengeCommon Mistakes
Leaving A negative in standard form — e.g., writing −2x + y = 5 instead of 2x − y = −5.
Standard form requires A to be greater than or equal to 0. If A is negative, multiply the entire equation by −1 to make it positive.
Forgetting to multiply every term when clearing fractions — e.g., multiplying only the fraction term.
When clearing fractions, multiply every term on both sides by the LCD. For y = (3/4)x + 2, multiply all three terms by 4: 4y = 3x + 8.
Confusing x-intercept and y-intercept: setting x = 0 to find the x-intercept.
x-intercept: set y = 0 (the line touches the x-axis, so y is zero). y-intercept: set x = 0 (the line touches the y-axis, so x is zero).
Thinking standard form and slope-intercept form describe different lines.
They describe the exact same line — just written differently. Converting between forms does not change the line, only its appearance on paper.
Math Tips
Standard form shortcut: the x-intercept is C/A and the y-intercept is C/B. For 3x + 4y = 12: x-int = 12/3 = 4, y-int = 12/4 = 3.
To convert slope-intercept to standard form: move the x-term left, then multiply by −1 if A is negative.
To clear fractions: find the LCD of all denominators, then multiply every term (on both sides) by the LCD.
Always check your conversion by substituting a point that satisfies the original equation into the new form.