Standard Form of a Line – Ax + By = C
Besides y = mx + c, a straight line can also be written in standard form: Ax + By = C, where A, B, and C are numbers (usually whole numbers, with A written positive by convention). Standard form treats x and y symmetrically – neither variable is singled out the way y is in y = mx + c – which makes it especially convenient for certain algebraic techniques, like solving two line equations simultaneously.
The two forms describe exactly the same line; they're just different ways of writing the same relationship, and you can always convert between them with a little rearranging.
Working with Standard Form
Standard form: Ax + By = C. To convert from y = mx + c, rearrange so both x and y terms are on one side. To find an intercept, set the other variable to 0 and solve.
Rearranging: y − 4x = 3, or equivalently 4x − y = −3.
Set y = 0: 2x + 5(0) = 20, so 2x = 20, giving x = 10. The x-intercept is (10, 0).
Real-Life Application
- Budget constraints: equations like “2 × apples + 3 × bananas = $12 budget” are naturally in standard form.
- Mixing problems: combining two ingredients to a fixed total is often written this way.
- Systems of equations: standard form is convenient when solving two equations together.
Key Takeaways
- Standard form is Ax + By = C, treating x and y symmetrically.
- It can always be converted to and from y = mx + c form.
- Setting one variable to 0 quickly finds the other axis's intercept.
Practice: Standard Form of a Line
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