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Grade 8 · unit review

Review: Systems of Two Linear Equations

A system requires the same ordered pair to satisfy every equation. Substitution and elimination find that shared pair, while parallel or coincident line structure explains whether a system has none or many solutions.

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Bring the unit skills together

Use these 4 new cases to check your reasoning. Choices are temporary and are not saved to an account, sent to staff or recorded as school grades. Download your current review choices before leaving or reloading.

Unit review questions
1. Solve y=2x+1 and x+y=22.
Need a hint?
  • Substitution uses one equation to express a variable, then replaces that variable in the other equation.
  • Use parentheses around a substituted expression, particularly when it is multiplied or subtracted.
2. Solve x+y=15 and x−y=-1.
Need a hint?
  • Elimination combines equations so one variable’s coefficients sum to zero.
  • Add entire equations, including their right sides.
3. Classify y=7x+1 and y=7x+3.
Need a hint?
  • Lines with equal slopes and different intercepts never meet in the coordinate plane.
  • Equal slopes alone do not settle the solution count.
4. A fictional stand sells $2 tickets and $3 tickets. It sells 15 tickets for $38. How many $3 tickets were sold?
Need a hint?
  • A total count and a weighted total can become two equations.
  • The counted groups must be disjoint and cover the whole supplied total.

The download contains your current choices and any checked feedback. Keep the separate transfer-task work in your own document; downloading its instructions does not mark it complete.

Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. Solve y=2x+1 and x+y=28. Candidate answers: (A) (10,19) (B) (19,9) (C) (9,19) (D) (9,20). 2. Solve x+y=19 and x−y=-1. Candidate answers: (A) (10,9) (B) (11,9) (C) (19,-1) (D) (9,10). 3. Classify y=9x+1 and y=9x+3. Candidate answers: (A) Infinitely many intersections (B) No intersection: parallel distinct lines (C) One intersection at x=0 (D) One intersection at x=9. 4. A fictional stand sells $2 tickets and $3 tickets. It sells 19 tickets for $48. How many $3 tickets were sold? Candidate answers: (A) 10 tickets (B) 11 tickets (C) 20 tickets (D) 9 tickets.

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. (9,19). Substitute 2x+1 for y: x+2x+1=28. Then 3x=27, x=9; y=2(9)+1=19. The ordered pair satisfies both equations. 2. (9,10). Add the equations so y cancels: 2x=18, x=9. Substitute in x+y=19 to get y=10. 3. No intersection: parallel distinct lines. The lines have equal slope 9 but different y-intercepts 1 and 3. Their vertical separation stays 2, so the system is inconsistent. 4. 10 tickets. If all 19 tickets were $2, revenue would be $38. Each $3 ticket adds $1. The extra $10 therefore means 10 of the $3 tickets.

A different response can also meet the criteria. Compare the evidence and reasoning, rather than copying the wording.