In the equation above, is a constant. For what value of does the equation have no solution?
A linear equation has no solution when its -terms cancel but its constants disagree. Distribute and collect the -terms, then use a Desmos regression to find the value that makes their coefficient zero. Check the constants afterward: canceling the -terms can also leave a statement that's true for every .
Hints
- Hint 1
Distribute the to both terms inside its parentheses. Then group the terms containing to find their coefficient, the number multiplying .
- Hint 2
For no solution, the -terms must cancel and leave a false statement, such as . What must the coefficient of equal for those terms to cancel?
- Hint 3
Use a regression in Desmos to find the value of that makes the coefficient zero. Then check the plain numbers to make sure they don't match.
Step-by-step
Cancel the x-terms, then check
Step 1Distribute the 5
Multiply by both terms inside its parentheses; keep the subtraction before :
- Step 2
Collect the x-terms
Take out of the two terms containing it. Their coefficient, the number multiplying , includes the whole :
- Step 3
Make the x-terms cancel
For no solution, the -terms must disappear and leave unequal plain numbers. If the coefficient of isn't zero, you can solve for one . Set it equal to zero:
- Step 4
Find k and check what remains
Type in Desmos. The subscript makes a value to find, and tells Desmos to fit it. Under PARAMETERS, Desmos shows . Substitute that value into the equation:
The last statement is false for every , so gives no solution. Choice C.