A system with no solution consists of two distinct parallel lines: they have the same slope but never meet. First collect the variable terms on one side of each equation. Then find the multiplier that matches their and coefficients, and check the constants. If the constants match too, the lines overlap instead. Use Desmos to confirm that your final lines stay apart.
Hints
- Hint 1
A coefficient is the number multiplying a variable. Before comparing coefficients, move the to the left side of the first equation. What is the new coefficient of ?
- Hint 2
For parallel lines, the and coefficients must change by the same multiplier. Once the first equation is simplified, what multiplier changes its coefficient to ?
- Hint 3
Matching variable coefficients isn't enough: check the constants. If they also change by the same multiplier, both equations describe the same line, which has infinitely many shared points.
Step-by-step
Match coefficients, then check constants
Step 1Collect the terms
Compare coefficients only after the terms are together. The first equation's coefficient won't stay , because there's also on the right. Subtract from both sides:
Combine the terms:
- Step 2
Reduce the first equation
Use smaller coefficients so the multiplier is easier to see. Divide every term by :
- Step 3
Match the variable coefficients
Parallel lines have the same slope. Match the coefficients first, then use that multiplier on the coefficients. The first line's becomes the second line's by doubling, so must double too:
- Step 4
Rule out the same line
Check the right-hand numbers: if they doubled too, both equations would describe the same line. Double the first line's constant:
The second constant is , not , so the lines are distinct and parallel. They won't meet.
- Step 5
Find and confirm
Multiply to find the constant:
Type and both original equations in Desmos. Their graphs stay separate, confirming that the system has no solution. The required constant is . Grid in -14.