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Grade 9 · unit review

Review: Algebra1: Systems and Feasible Linear Models

A mathematical plan can require simultaneous equalities or inequalities. The feasible set contains inputs satisfying all constraints, and a linear objective asks which allowed choice gives the greatest or least modeled value.

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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 2x+y=22 and x−y=-1.
Need a hint?
  • A system can be solved by adding equations whose opposite variable coefficients cancel.
  • The operation applies to right-side constants as well as coefficients.
2. Which point satisfies x≥0, y≥0 and x+y≤14?
Need a hint?
  • A feasible point satisfies all listed restrictions at once.
  • Passing one inequality does not make a point feasible.
3. How should y<2x+7 be represented on a coordinate plane?
Need a hint?
  • A strict linear inequality excludes its equality line, represented by a dashed boundary.
  • A solid line includes equality, so it would misrepresent a strict sign.
4. For x≥0, y≥0, x+y≤10, maximize P=2x+3y.
Need a hint?
  • An objective assigns a modeled value to each feasible choice.
  • An optimum must satisfy the original constraints, not merely make the expression large.

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 2x+y=28 and x−y=-1. Candidate answers: (A) (10,9) (B) (11,9) (C) (28,-1) (D) (9,10). 2. Which point satisfies x≥0, y≥0 and x+y≤18? Candidate answers: (A) (10,9) (B) (9,9) (C) (9,−1) (D) (−1,9). 3. How should y<2x+9 be represented on a coordinate plane? Candidate answers: (A) Dashed boundary line; shade above it (B) Dashed boundary line; shade below it (C) Solid boundary line; shade above it (D) Solid boundary line; shade below it. 4. For x≥0, y≥0, x+y≤12, maximize P=2x+3y. Candidate answers: (A) 35 (B) 36 (C) 37 (D) 72.

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. (9,10). Add the equations: 3x=27. Thus x=9, and y=x−(-1)=10. Verification in the first equation gives 28. 2. (9,9). The point (9,9) has nonnegative coordinates and sum 18, equal to the inclusive cap. The first distractor exceeds the cap; the others use a negative coordinate. 3. Dashed boundary line; shade below it. The strict sign excludes equality, so the boundary is dashed. Smaller y-values lie below the line y=2x+9. Test any convenient off-line point to check the side. 4. 36. Each unit assigned to y contributes 3 rather than 2 to P, so the maximum uses x=0,y=12. P=36. Equivalently P=2(x+y)+y≤2(12)+12.

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