Solve the system x + y = 1/12 and 8x + 15y = 1.

Step-by-step solution

1. 1

   Rearrange the simpler equation. The first equation has unit coefficients, so isolate $x$ there:

   $x=\frac{1}{12}-y$

   Keeping the fractions exact (rather than writing $0.0833$) is what makes the clean answer visible at the end.

2. 2

   Substitute into the second equation.

   $8\left(\frac{1}{12}-y\right)+15y=1\quad\Rightarrow\quad \frac{8}{12}-8y+15y=1$

3. 3

   Simplify. Since $\frac{8}{12}=\frac23$ and $-8y+15y=7y$,

   $\frac23+7y=1\quad\Rightarrow\quad 7y=1-\frac23=\frac13\quad\Rightarrow\quad y=\frac{1}{21}$

4. 4

   Back-substitute for $x$. Using a common denominator of $84$:

   $x=\frac{1}{12}-\frac{1}{21}=\frac{7}{84}-\frac{4}{84}=\frac{3}{84}=\frac{1}{28}$

5. 5

   Check both equations. First: $\frac{1}{28}+\frac{1}{21}=\frac{3}{84}+\frac{4}{84}=\frac{7}{84}=\frac{1}{12}$ ✓. Second: $\frac{8}{28}+\frac{15}{21}=\frac27+\frac57=1$ ✓.

6. 6

   Interpret the fractions. Reading $x$ and $y$ as work rates (jobs per day), the first equation says that working together they need $12$ days, and the second says $8$ days of the first worker plus $15$ days of the second completes one job. The answer then means the two would need $28$ and $21$ days respectively on their own.