A linear system with a missing coefficient may ask how two lines relate rather than where they cross. No solution means distinct parallel lines: one multiplier matches both variable coefficients, but not the constants. Match the variable coefficients first, then check the constants. If the constants match too, the equations describe the same line.
Hints
- Hint 1
For two lines, no solution means they never meet. They must be parallel, pointing in the same direction without lying on top of each other. Which pair of coefficients lets you find their multiplier without knowing ?
- Hint 2
The coefficients go from to , giving a multiplier of . Apply that multiplier to the first equation's term. Does it also turn the first constant, , into ?
Step-by-step
Match the coefficients
Step 1Find the multiplier from the terms
A solution is a point on both lines. No solution means the lines must be separate and parallel, so their variable coefficients need the same multiplier. The coefficients go from to . Divide the second by the first:
- Step 2
Apply the multiplier to the variable terms
Multiply the whole left side of the first equation by :
The second equation already has , so its coefficient must match the one in this scaled expression for the lines to be parallel.
- Step 3
Check that the lines are different
The constants are the numbers without variables. Multiply the first constant by the same number:
Compare it with the second constant:
The constants differ, so the parallel lines are not the same line. If they matched, the system would have infinitely many solutions instead.
- Step 4
Read and check the graph
The coefficient in is , so type and then the two given equations on separate lines in Desmos. The graph shows two distinct parallel lines. They never meet, so the system has no solution when . Choice C.