Standard Form of a Line: notes and common mistakes

Key ideas

• Standard form is Ax + By = C, with A and B not both 0. The graph is a straight line.
• The usual convention is integers for A, B and C, with A not negative and no common factor. It is a convention, not a law of math.
• Cover the y term to find the x-intercept C/A (if A ≠ 0). Cover the x term to find the y-intercept C/B (if B ≠ 0).
• To get slope-intercept form, solve for y. The slope is −A/B and the y-intercept is C/B (if B ≠ 0).
• A = 0 gives a horizontal line. B = 0 gives a vertical line, which standard form can write but slope-intercept form cannot.
• Standard form fits totals of two quantities: price × number for each item, added to a fixed total.

Steps

1. To find intercepts: cover the y term and solve for x, then cover the x term and solve for y.
2. To graph: plot the two intercept points, draw the line through them, and check a third point.
3. To reach slope-intercept form: subtract the x term, then divide every term by B.
4. To reach standard form: clear fractions, move the x and y terms to the left and the number to the right, make A not negative, and remove any common factor.

Formulas

• Standard form: Ax + By = C, with A and B not both 0
• x-intercept: x = C/A (if A ≠ 0), so the line crosses at (C/A, 0)
• y-intercept: y = C/B (if B ≠ 0), so the line crosses at (0, C/B)
• Slope: m = −A/B (if B ≠ 0)
• Horizontal line: A = 0: y = C/B, slope 0
• Vertical line: B = 0: x = C/A, slope undefined

Watch out

• Signs belong to their numbers: in 2x − 5y = 10, B = −5.
• The slope is −A/B, with a minus sign. It is not A/B.
• To find the x-intercept, set y = 0 (cover the y term). Setting x = 0 finds the y-intercept.
• When you divide by a negative B, every term changes sign. Divide every term, not just one.
• If C = 0, both intercepts are 0 and the line crosses both axes at (0, 0), so you need another point.

Why it works

• Why cover a term? The x-axis has y = 0, so the y term becomes 0 and disappears. The y-axis has x = 0, so the x term disappears.
• Why is the slope −A/B? Solving Ax + By = C for y gives y = −(A/B)x + C/B, and the number multiplying x is the slope.
• Why can you multiply every term by a number that is not 0? Doing the same thing to both sides keeps the same solutions, so the graph does not change. Multiplying by 0 would destroy the equation.