Question

solve the equation. remember to check for extraneous solutions.
\\(\frac{1}{k - 2}=\frac{4k + 6}{k^{2}-2k}\\)
\\(k = \frac{7}{4}\\)
\\(k=-\frac{5}{4}\\)
\\(k = - 2\\)
\\(k = 2\\)
question #12
eleven liters of a 26% alcohol solution was mixed with seven liters of an 8% alcohol solution. what is the concentration of the mixture? round your answer to the nearest hundredth.
12%
33%
19%
10%

Explanation:

Step1: Factor the denominator

Factor $k^{2}-2k$ as $k(k - 2)$. The equation $\frac{1}{k - 2}=\frac{4k + 6}{k^{2}-2k}$ becomes $\frac{1}{k - 2}=\frac{4k+6}{k(k - 2)}$.

Step2: Multiply both sides by $k(k - 2)$

Since $k
eq0$ and $k
eq2$ (to avoid division - by - zero), we have $k=4k + 6$.

Step3: Solve for $k$

Subtract $4k$ from both sides: $k-4k=4k + 6-4k$, which gives $-3k=6$. Then divide both sides by $-3$: $k=-2$.

Step4: Check for extraneous solutions

The original equation has denominators $k - 2$ and $k^{2}-2k$. When $k=-2$, $k-2=-2 - 2=-4
eq0$ and $k^{2}-2k=(-2)^{2}-2\times(-2)=4 + 4 = 8
eq0$. So $k=-2$ is a valid solution.

for second question:

Step1: Calculate the amount of alcohol in each solution

For the $26\%$ alcohol solution with $11$ liters, the amount of alcohol is $0.26\times11 = 2.86$ liters. For the $8\%$ alcohol solution with $7$ liters, the amount of alcohol is $0.08\times7=0.56$ liters.

Step2: Calculate the total amount of alcohol and total volume of the mixture

The total amount of alcohol is $2.86 + 0.56=3.42$ liters. The total volume of the mixture is $11 + 7 = 18$ liters.

Step3: Calculate the concentration of the mixture

The concentration $C$ of the mixture is $C=\frac{3.42}{18}=0.19 = 19\%$

Answer:

$k=-2$