Linear systems and graph intersections
Combine two constraints and distinguish a single intersection, parallel lines and the same line.
What you will tackle
- Create and interpret simultaneous contextual constraints
- Solve by substitution, elimination or structure
- Connect algebraic solutions to graph intersections
- Find parameter conditions for zero, one or infinitely many solutions
Both equations must hold
A solution of a system satisfies every equation at once. For x + y = 9 and 2x − y = 6, adding eliminates y: 3x = 15. The solution (5,4) is the intersection, not two separate answers chosen from different lines.
Proportional coefficients need a constant check
If one entire equation is a nonzero multiple of the other, both describe one line and there are infinitely many solutions. If just the x and y coefficients have that proportion while the constants do not, the lines are parallel and there is no solution. Otherwise two nonparallel lines meet once.
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
Solve x + y = 9 and 2x − y = 6; explain the graphical meaning.
Show a starting hint
Adding the equations eliminates y.
Compare with the explained answer
- Add the equations to obtain 3x = 15, so x = 5.
- Use x + y = 9 to get y = 4.
- Substitute into both equations; the graphs intersect at (5,4).
Answer / response criteria: (5,4), the single intersection.
Independent practice
Independent 1
Classify x + 2y = 5 together with 3x + 6y = k. Give the condition on k.
Compare with the explained answer
- Multiply the first equation by 3: 3x + 6y = 15.
- If k = 15, the equations are equivalent.
- If k ≠ 15, the same left side would have to equal two different constants.
Answer / response criteria: Infinitely many solutions when k = 15; no solution otherwise.
Application practice
Application 1
A cinema sold 30 tickets: adults $11 and students $7. Total receipts were $266. Find both counts and verify both constraints.
Compare with the explained answer
- Write a + s = 30 and 11a + 7s = 266.
- Subtract 7 times the count equation: 4a = 56, so a = 14.
- Then s = 16; 14 + 16 = 30 and 154 + 112 = 266.
Answer / response criteria: 14 adults and 16 students.
What can you explain now?
A cinema sold 30 tickets: adults $11 and students $7. Total receipts were $266. Find both counts and verify both constraints.
Compare with the explanation
14 adults and 16 students.
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