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Grade 8 · unit review

Review: Linear Equations: One, Many, or No Solutions

An equation’s solution count depends on its entire structure. Balanced simplification may isolate a variable, reduce to an identity, or expose a contradiction; each outcome has a different interpretation.

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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 5x+8=2x+29.
Need a hint?
  • When a variable appears on both sides, subtract a matching variable term from both sides before isolating the remaining coefficient.
  • Moving a term is shorthand for an operation on both sides, not permission to change its sign arbitrarily.
2. How many real solutions does 7(x+2)=7x+14 have?
Need a hint?
  • Equivalent left and right expressions produce a true equality for every allowed input.
  • Cancellation does not automatically mean no solution.
3. How many real solutions does 7x+8=7x+9 have?
Need a hint?
  • Equal variable coefficients can cancel even when the equation is false for every input.
  • Do not divide by a zero coefficient to invent a numerical answer.
4. Solve x/2+x/3=35.
Need a hint?
  • Multiplying both sides by the least common denominator can turn fractional coefficients into whole-number coefficients.
  • An equation multiplier applies to the whole side, not merely the fractions that look inconvenient.

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 5x+10=2x+37. Candidate answers: (A) 10 (B) 18 (C) 8 (D) 9. 2. How many real solutions does 9(x+2)=9x+18 have? Candidate answers: (A) Every real x (B) Exactly one, x=2 (C) Exactly two, x=0 and x=2 (D) No real x. 3. How many real solutions does 9x+10=9x+11 have? Candidate answers: (A) Every real x (B) Exactly one, x=1 (C) Exactly one, x=10 (D) No real x. 4. Solve x/2+x/3=45. Candidate answers: (A) 108 (B) 53 (C) 54 (D) 55.

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. Subtract 2x and then 10 from both sides: 3x=27. Divide by 3 to get x=9. Both original sides equal 55. 2. Every real x. Distribute the left side: 9x+18. Both sides are identical, so every real value satisfies the equation. No division by a zero coefficient is needed. 3. No real x. Subtract 9x from both sides. This leaves 10=11, which is false. A contradiction means the original equation has no solution. 4. 54. The least common denominator is 6. Multiply every term by 6: 3x+2x=270. Thus 5x=270 and x=54.

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