To solve the system of equations by the elimination method, follow these steps:

The equations provided are:

1. $\left(-\frac{1}{20} - s \right)y_1 + \frac{5}{18}y_2 = -\frac{1 - 11}{5}$
2. $\frac{5}{18}y_1 + \left(-\frac{5}{18} - 5 \right)y_2 = -7$

Step 1: Rewrite the system for simplicity

Let us simplify the constants where possible.

For the first equation: $\text{Right-hand side: } -\frac{1-11}{5} = -\frac{-10}{5} = 2$ So, equation (1) becomes: $\left(-\frac{1}{20} - s \right)y_1 + \frac{5}{18}y_2 = 2$

The second equation remains: $\frac{5}{18}y_1 + \left(-\frac{5}{18} - 5 \right)y_2 = -7$

Step 2: Align the coefficients for elimination

To eliminate one variable, align the coefficients of $y_1$ or $y_2$. Let's eliminate $y_2$.

The coefficients of $y_2$ in the two equations are $\frac{5}{18}$ and $-\frac{5}{18} - 5$. To eliminate $y_2$, we will multiply both equations to match these coefficients.

Multiply the first equation by $-\left(-\frac{5}{18} - 5\right)$:

$\left(-\frac{1}{20} - s \right)y_1 \cdot \left(-\frac{5}{18} - 5\right) + \frac{5}{18}y_2 \cdot \left(-\frac{5}{18} - 5\right) = 2 \cdot \left(-\frac{5}{18} - 5\right)$

Multiply the second equation by $\frac{5}{18}$:

$\frac{5}{18} \cdot \frac{5}{18}y_1 + \left(-\frac{5}{18} - 5\right) \cdot \frac{5}{18}y_2 = -7 \cdot \frac{5}{18}$

Step 3: Solve for $y_1$

After eliminating $y_2$, solve for $y_1$. Substitute $y_1$ into one of the original equations to find $y_2$.

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