Question

Solve the equation. Check your solution.
one fifth
1
5​(minus−1010kplus+5050​)equals=77plus+1212k

Answer 1

To solve the equation given as:

\[
\frac{1}{5}(-10k + 50) = 77 + 12k
\]

we will start by eliminating the fraction on the left side. We can do this by multiplying both sides of the equation by 5:

\[
-10k + 50 = 5(77 + 12k)
\]

Calculating the right side:

\[
5(77 + 12k) = 385 + 60k
\]

Now we rewrite the equation:

\[
-10k + 50 = 385 + 60k
\]

Next, we can move all terms involving \( k \) to one side and constant terms to the other side. Adding \( 10k \) to both sides:

\[
50 = 385 + 60k + 10k
\]
\[
50 = 385 + 70k
\]

Now, subtract 385 from both sides:

\[
50 - 385 = 70k
\]
\[
-335 = 70k
\]

To isolate \( k \), divide both sides by 70:

\[
k = \frac{-335}{70}
\]

Now we can simplify \( \frac{-335}{70} \):

\[
k = \frac{-335 \div 5}{70 \div 5} = \frac{-67}{14}
\]

Thus, the solution for \( k \) is:

\[
k = -\frac{67}{14}
\]

### Check the Solution

We'll substitute \( k = -\frac{67}{14} \) back into the original equation to check if both sides are equal.

First, calculate the left side:

\[
-10k + 50 = -10\left(-\frac{67}{14}\right) + 50 = \frac{670}{14} + 50
\]

Convert 50 to a fraction with a common denominator:

\[
50 = \frac{700}{14}
\]
\[
-10k + 50 = \frac{670 + 700}{14} = \frac{1370}{14}
\]

Now, calculate the left side:

\[
\frac{1}{5}\left(-10k + 50\right) = \frac{1}{5}\left(\frac{1370}{14}\right) = \frac{1370}{70} = \frac{137}{7}
\]

Now calculate the right side:

\[
77 + 12k = 77 + 12\left(-\frac{67}{14}\right) = 77 - \frac{804}{14} = 77 - \frac{804}{14}
\]
Convert 77 to a fraction with a common denominator:

\[
77 = \frac{1078}{14}
\]
Then,

\[
77 + 12k = \frac{1078 - 804}{14} = \frac{274}{14} = \frac{137}{7}
\]

Both sides yielded \( \frac{137}{7} \), so the solution is confirmed correct:

\[
\boxed{-\frac{67}{14}}
\]