Choose a representation
Compare coefficient ratios and constants separately. For a line meeting a parabola, substitute the line expression, solve the resulting equation and recover the requested coordinate.
Worked example
The system 4 x+2 y=7 and kx+6 y=22 has no solution. What is k? No solution requires parallel lines with proportional variable coefficients. Match the y-coefficient factor 3; constants 7×3 and 22 differ. Substitute the given numbers: 4 × (6 ÷ 2) = 12. The requested value is 12.
Contrast a wrong operation
In that example, 4 follows this error: Match the x coefficients without matching the y scaling. The correct requested value is 12. Explain which quantity the mistaken method is actually calculating.
Connect a second form
The graphs y=x² and y=16 intersect. What is the distance between their two x-coordinates? Set x² equal to the constant output to obtain positive and negative roots. Subtract the negative x-coordinate from the positive one. Substitute the given numbers: 2 × 4 = 8. The requested value is 8.
Transfer and check
The system 5 x+2 y=7 and kx+6 y=22 has no solution. What is k? Pause before checking. Answer: 15. No solution requires parallel lines with proportional variable coefficients. Match the y-coefficient factor 3; constants 7×3 and 22 differ. Substitute the given numbers: 5 × (6 ÷ 2) = 15.
Boundaries of the method
A zero determinant means the variable coefficients are dependent; constants decide whether the equations agree or contradict. Do not count a repeated root as two distinct points. This lesson samples a PSAT/NMSQT objective; its examples are original and do not claim official question status or full coverage.
The system 4 x+2 y=7 and kx+6 y=22 has no solution. What is k?
- No solution requires parallel lines with proportional variable coefficients.
- Match the y-coefficient factor 3; constants 7×3 and 22 differ.
- Substitute the given numbers: 4 × (6 ÷ 2) = 12.
- The requested value is 12.
Compare the other answer choices
Match the x coefficients without matching the y scaling.
Multiply by the second y coefficient instead of the ratio of y coefficients.
Match the constant ratio instead of the y-coefficient ratio: 4×22/7.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
Math specifications