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Let's solve the system of equations step by step. The given system of equations is:

\[
2y - 19x = -1 \quad \text{(1)}
\]
\[
8y + 19x = 91 \quad \text{(2)}
\]

### Step 1: Add the two equations

First, add the two equations together to eliminate \( x \):

\[
(2y - 19x) + (8y + 19x) = -1 + 91
\]

This simplifies to:

\[
10y = 90
\]

### Step 2: Solve for \( y \)

Now, divide both sides of the equation by 10:

\[
y = \frac{90}{10} = 9
\]

### Step 3: Substitute \( y = 9 \) into one of the original equations

Substitute \( y = 9 \) into equation (1):

\[
2(9) - 19x = -1
\]

Simplify:

\[
18 - 19x = -1
\]

Now, subtract 18 from both sides:

\[
-19x = -1 - 18 = -19
\]

Finally, divide both sides by -19:

\[
x = \frac{-19}{-19} = 1
\]

### Final solution:

\[
x = 1, \quad y = 9
\]

Would you like further explanation on any step? Here are five related questions:

1. Can you solve a system of equations using substitution instead of elimination?
2. How can you solve a system of equations graphically?
3. What happens if a system of equations has no solution or infinitely many solutions?
4. How would you check the solution of a system of equations?
5. Can you solve this system using matrix methods (like Gaussian elimination)?

**Tip:** Always double-check your solution by substituting the values back into the original equations.

Solve the system of equations: 2y - 19x = -1 and 8y + 19x = 91

Learn how to solve the system of linear equations 2y - 19x = -1 and 8y + 19x = 91 using the elimination method. This problem covers key algebraic concepts, including variable elimination and substitution. Suitable for students in Grades 9-12.

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