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
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide