A system of two lines has no solution when the lines are parallel, so they never meet. Match both variable coefficients using one multiplier, then check that the constants do not match. Start with the coefficients that contain no unknown constant. Matching the variable coefficients alone is not enough: the equations could describe the same line.
Hints
- Hint 1
For no solution, the lines must be parallel. Their and coefficients must scale by the same number. Which pair of coefficients lets you find that number without using ?
- Hint 2
The coefficients give a multiplier of . Apply it to , the first equation’s coefficient, and match the second equation’s coefficient.
- Hint 3
Matching the variable coefficients might also make the equations describe the same line. Compare the numbers on the right using the multiplier to distinguish one line from two parallel lines.
Step-by-step
Match coefficients, then check the constants
Step 1Find the multiplier from the y coefficients
A coefficient is the number multiplying a variable. The first coefficient is , and the second is , so the multiplier is positive :
- Step 2
Match the x coefficients
For the lines to be parallel, that same must turn the first coefficient into the second. The second is positive , so:
- Step 3
Divide to isolate the parentheses
Divide both sides by to find the first coefficient:
- Step 4
Check that the lines are distinct
The constant terms are the numbers on the right. Multiplying the first equation’s constant by does not give the second equation’s constant: . So the matching variable coefficients give parallel lines, not the same line.
- Step 5
Find k
Type in Desmos. Use for the unknown and to have Desmos find its value. Under PARAMETERS, Desmos shows . So the system has no solution when . Grid in 1.