A system of linear equations has no solution when its lines are parallel but different. Match the and coefficients using one multiplier, then check that the constants do not match under that multiplier. Desmos can confirm that the lines never meet. If the constants match too, the equations describe the same line instead.
Hints
- Hint 1
For two lines to have no shared point, their and coefficients must change by the same multiplier. What number turns the first equation’s into the second equation’s ?
- Hint 2
Matching the coefficients gives the lines the same direction, but they could still be the same line. Multiply the first equation’s by your multiplier. Does it become ?
- Hint 3
The missing coefficient must use that same multiplier. Start with the first equation’s , not its constant . What must multiply in the second equation?
Step-by-step
Match coefficients, then check constants
Step 1Find the coefficient multiplier
A coefficient is the number multiplying a variable. The coefficients go from to , so the multiplier is : . For the lines to be parallel, their coefficients must use that same multiplier.
- Step 2
Check that the lines are different
Apply the multiplier to the first equation’s constant: . The second equation has , not . So if its coefficient matches, the lines will be parallel, meaning they never meet, rather than the same line. Matching coefficients but different constants means no solution.
- Step 3
Find and confirm in Desmos
The first coefficient is , so use the same multiplier to find . Type in Desmos, then type the two given equations on separate lines. Desmos shows and two lines that never meet. The system has no solution when . Choice B.