General-form lines and contextual models
| English | 中文 | Pinyin |
|---|---|---|
| general form/ˈdʒenərəl fɔːm/ | 一般式 | yì bān shì |
Which part of a delivery charge changes with distance?
- A straight-line cost model has a fixed charge and a rate per kilometre. Its intercept and gradient have different meanings and units.
- This lesson studies general form 一般式: The form ax+by+c=0 for a straight line, with a and b not both zero.
Choose the mathematical structure
- For a nonvertical line through two points, m=(y₂−y₁)/(x₂−x₁) and y−y₁=m(x−x₁). Rearrange into ax+by+c=0. Parallel nonvertical lines have equal gradients; perpendicular finite gradients multiply to −1. Treat vertical and horizontal lines separately.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines general form?
The form ax+by+c=0 for a straight line, with a and b not both zero.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Through A(−1,3) and B(5,−1), m=−4/6=−2/3. Thus y−3=−2(x+1)/3, giving 2x+3y−7=0. The perpendicular through A has gradient 3/2 and equation 3x−2y+9=0. A vertical line x=4 is perpendicular to a horizontal line y=3 without having a finite gradient. Delivery quotes C=8 at d=0 km and C=38 at d=12 km give rate (38−8)/12=2.5 currency units per kilometre and C=8+2.5d. At d=8, the predicted cost is 28.
General-form lines and contextual models
For a nonvertical line through two points, m=(y₂−y₁)/(x₂−x₁) and y−y₁=m(x−x₁)
Match each coordinate calculation to its geometric or contextual condition.
Find the gradient of the line through (−1,3) and (5,−1).
m=(−1−3)/(5−(−1))=−4/6=−2/3.
Test a tempting shortcut
- For ax+by+c=0, the gradient is −a/b only if b≠0. Do not confuse the fixed charge with the price per kilometre. A line fitted to two quoted distances does not prove the same tariff applies at all distances.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every straight line has a finite gradient and can be written as y=mx+c. This claim is false. Explain which definition or assumption it violates.
Find the gradient of its nonvertical perpendicular.
m perpendicular=−1/(−2/3)=3/2.
Every straight line has a finite gradient and can be written as y=mx+c.
For ax+by+c=0, the gradient is −a/b only if b≠0. Do not confuse the fixed charge with the price per kilometre. A line fitted to two quoted distances does not prove the same tariff applies at all distances.
Interpret a new situation
- Interpret the cost intercept, gradient and nonnegative distance domain. Within 0≤d≤12, interpolation uses the observed quote range; beyond it, extrapolation needs a tariff assumption. Check both original points after rearranging a geometric line.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For C=8+2.5d, find the predicted cost at d=8.
C(8)=8+2.5×8=28.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · C. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The form ax+by+c=0 for a straight line, with a and b not both zero. Choose the relationship, show the method, check its assumptions and interpret the result.