A linear system with no solution draws two distinct parallel lines. Match the - and -coefficients by one multiplier, then use the extra condition to find possible values of the constants. Check the numbers on the right too: if they scale by the same multiplier, the lines overlap and have infinitely many solutions. Desmos can confirm which pair makes two separate lines.
Hints
- Hint 1
Lines with no intersection are parallel: their - and -coefficients scale together. Compare the coefficient pairs and . What product equation expresses that scaling?
- Hint 2
Use the given sum together with the product you found. Which two numbers add to and multiply to ? You still need to decide which number is .
- Hint 3
Sharing a slope doesn't guarantee no solution. If one whole equation, including its right-hand side, is a multiple of the other, the lines coincide: every point on either line solves both. Check the two possible assignments.
Step-by-step
Match coefficients, then check the constants
Step 1Find the condition for parallel lines
No solution means the two lines must be parallel. Their coefficient pairs, and , must scale by the same multiplier. Equate their cross-products to express that condition:
- Step 2
Find the two possible assignments
The given sum is . Together with , this makes and equal to and in some order: and . Don't choose yet; the order matters.
- Step 3
Rule out the overlapping lines
Try and . The first equation becomes:
Multiply the entire equation by :
That's exactly the second equation. These lines coincide, so they have infinitely many solutions, not none.
- Step 4
Confirm the remaining pair in Desmos
For the remaining order, type , , , and on separate Desmos lines. Desmos shows two distinct parallel lines. Both left sides become , but one equals and the other equals , so the lines cannot meet. The value of is . Choice B.