Connect two-variable equations, tables and lines
Interpret pairs that satisfy a linear constraint and construct a line from points or slope information.
What you will tackle
- Model constraints and interpret coefficients and solution pairs
- Connect equations, tables and graphical lines inside and outside contexts
- Find one quantity from another
- Construct lines from two points, a point and slope, or parallel/perpendicular information
Pairs are solutions
The line 2x + y = 10 contains every pair satisfying the constraint. Setting x = 0, 2 and 5 gives y = 10, 6 and 0. Plotting those pairs gives the same line as y = −2x + 10. A point lies on the line only if substituting both coordinates makes the equation true.
Build a rule from geometric information
For distinct points, slope is change in y divided by change in x. Through (x₁,y₁) with slope m, use y − y₁ = m(x − x₁). Distinct parallel nonvertical lines share a slope; perpendicular nonvertical lines have slopes whose product is −1. Vertical lines have x constant and are perpendicular to horizontal lines.
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
Find a line through (1,5) and (4,11); give its y-intercept and a third point.
Show a starting hint
Use the difference in outputs over the difference in inputs.
Compare with the explained answer
- The slope is (11 − 5)/(4 − 1) = 2.
- Use y − 5 = 2(x − 1), giving y = 2x + 3.
- The intercept is (0,3); substituting x = 2 gives (2,7).
Answer / response criteria: y = 2x + 3; intercept (0,3); for example (2,7).
Independent practice
Independent 1
Through (2,3), find a line parallel to y = 4x − 1 and a line perpendicular to it. Does (3,7) lie on the parallel line?
Compare with the explained answer
- The parallel slope is 4, so y − 3 = 4(x − 2), or y = 4x − 5.
- The perpendicular slope is −1/4, so y − 3 = −(x − 2)/4.
- At x = 3 the parallel line gives y = 7.
Answer / response criteria: Parallel: y = 4x − 5. Perpendicular: y = −x/4 + 7/2. Yes, (3,7) lies on the parallel line.
Application practice
Application 1
An equipment budget satisfies 12x + 8y = 96, where x and y are counts of two items. Interpret the intercepts, find y when x = 4, and explain the meaningful part of the graph.
Compare with the explained answer
- With y = 0, x = 8; with x = 0, y = 12.
- For x = 4, 48 + 8y = 96, so y = 6.
- Only nonnegative whole-number pairs are actual purchase plans, although the equation defines an entire real line.
Answer / response criteria: Intercepts (8,0) and (0,12); (4,6) is a purchase plan; only nonnegative integer pairs are meaningful.
What can you explain now?
An equipment budget satisfies 12x + 8y = 96, where x and y are counts of two items. Interpret the intercepts, find y when x = 4, and explain the meaningful part of the graph.
Compare with the explanation
Intercepts (8,0) and (0,12); (4,6) is a purchase plan; only nonnegative integer pairs are meaningful.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
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Source and review status
Core coverage follows the detailed SAT skills in Appendix B of the September 2026 framework. The conditional School Day Essay is separate. The framework is maintained rather than issued as an annual subject syllabus.
Original teaching and explained self-check practice are available. Independent subject-specialist review and difficulty calibration remain pending; this is not an exam-board endorsement or a mastery certificate.
Official source specification / guide