In the equation above, is a constant. For what value of does the equation have no solutions?
For a linear equation, a question about no solutions asks you to make the -terms cancel without making both sides identical. Match the -coefficients, then check that the constants differ. Distribute to expose those parts, and graph both sides in Desmos to confirm they never meet. Matching the coefficients alone could instead give infinitely many solutions.
Hints
- Hint 1
A linear equation has only to the first power. For it to have no solutions, the -terms must cancel while the plain numbers disagree. After distributing , what multiplies on each side?
- Hint 2
A coefficient is what multiplies . Set the two coefficients equal and solve for . Don't stop there: if the plain numbers also match, every value of works.
- Hint 3
The constant terms are the parts without . Substitute your value of into both of them. Are they different, as a no-solution equation requires?
Step-by-step
Match coefficients, then check constants
Step 1Expose the parts of each side
For no solutions, the -terms must cancel but the remaining numbers must disagree. Distribute on the left so you can compare those parts:
- Step 2
Match the coefficients of
A coefficient is what multiplies : it's on the left and on the right. To make the -terms cancel, set those coefficients equal:
- Step 3
Find the value of
Divide both sides by : . This is not yet ; first use it to check the parts without .
- Step 4
Check the constant terms
The constant terms are and . Substitute and multiply: . The -terms cancel, but the constants differ, so no can make the equation true.
- Step 5
Find and confirm
Add to both sides of : . In Desmos, type , then graph each side as and . It shows parallel lines, and , with no intersection. The requested value of is . Grid in 5.