Find When a Coefficient Creates No Solution
A parameter controls an equation’s structure.
About 20 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: For what real k does kx+2=4x+3 have no solution? Candidate answers: (A) 0 (B) 3 (C) 4 (D) 5.
- Independent check inputs: For what real k does kx+3=5x+4 have no solution? Candidate answers: (A) 0 (B) 4 (C) 5 (D) 6.
Learn the idea
An equation may describe distance, a bounded interval, a parameter-dependent relation or a formula between several quantities. Its solution must respect all stated conditions, including endpoint inclusion and nonzero divisors. A parameter controls an equation’s structure. Making the variable coefficients equal can remove the variable entirely, after which the remaining constants determine identity or contradiction.
Here is the method on the supplied teaching case. At k=4, the variable coefficients cancel and leave 2=3, false. At any other k, a nonzero coefficient gives one solution.
Do not confuse a parameter value with a variable solution. First identify when the variable coefficient vanishes; then examine the constant statement rather than dividing by that zero coefficient.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: For what real k does kx+2=4x+3 have no solution? Candidate answers: (A) 0 (B) 3 (C) 4 (D) 5. Solution: At k=4, the variable coefficients cancel and leave 2=3, false. At any other k, a nonzero coefficient gives one solution. The final answer is 4. Do not confuse a parameter value with a variable solution. First identify when the variable coefficient vanishes; then examine the constant statement rather than dividing by that zero coefficient.
Check your reasoning
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Helpful tips
- A parameter controls an equation’s structure.
- Do not confuse a parameter value with a variable solution.
See the answer and explanation
5
At k=5, the variable coefficients cancel and leave 3=4, false. At any other k, a nonzero coefficient gives one solution.
Take it further
Changed-input task: For what real k does kx+9=11x+10 have no solution? Candidate answers: (A) 0 (B) 10 (C) 11 (D) 12. Show the mathematical reason, not only the selected answer. Expected result: 11. At k=11, the variable coefficients cancel and leave 9=10, false. At any other k, a nonzero coefficient gives one solution.
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More practice: try new cases
Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.
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Check the subject sources
- OpenStax: Linear inequalities and absolute value inequalities
- OpenStax: Linear equations in one variable
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