In a linear system, no solution means the two lines are parallel but not the same line. Find the multiplier that makes their and coefficients match, then check the numbers on the right. If those numbers match too, the lines overlap and have infinitely many solutions. Desmos can confirm that the final pair never crosses.
Hints
- Hint 1
Two distinct parallel lines never meet. For no solution, look for one multiplier that takes both variable coefficients in the first equation to those in the second.
- Hint 2
Start with the coefficients because neither contains . What multiplies to give ? Keep both minus signs when you compare them.
- Hint 3
After matching the variable coefficients, check the numbers on the right. If they also match under the same multiplier, the equations describe one line with infinitely many solutions.
Step-by-step
Match coefficients, then check the constants
Step 1Find the multiplier from the y terms
A solution is a point shared by both lines. For no solution, the lines must be distinct and parallel, so both variable coefficients need the same multiplier. The terms tell you that multiplier:
- Step 2
Scale the first equation
Multiply both sides of the first equation by :
Multiply each term:
- Step 3
Check that the lines will stay separate
The second equation has on the right, not . So if its coefficient matches the scaled equation too, the lines will be parallel but won't overlap. For no solution, the variable coefficients must scale together, but the constants must not.
- Step 4
Set the x coefficients to match
The second equation's coefficient is . Give it the same multiplier as the coefficient:
- Step 5
Evaluate and confirm in Desmos
Type , then type both given equations. Desmos shows and two separate parallel lines, so they never meet. The system has no solution when . Choice C.