Solve the given linear equation: 2x-7y=7; 3x+y=22.

To solve the system of linear equations:

1. \( 2x - 7y = 7 \)
2. \( 3x + y = 22 \)

We will use the substitution or elimination method. Here, we use the substitution method:

Step 1: Solve one equation for one variable
From equation (2): \( 3x + y = 22 \), solve for \( y \):
\( y = 22 - 3x \)

Step 2: Substitute \( y = 22 - 3x \) into equation (1):
Substitute into \( 2x - 7y = 7 \):
\( 2x - 7(22 - 3x) = 7 \)
Expand the equation:
\( 2x - 154 + 21x = 7 \)
Combine like terms:
\( 23x - 154 = 7 \)
Add 154 to both sides:
\( 23x = 161 \)
Solve for \( x \):
\( x = \frac{161}{23} = 7 \)

Step 3: Substitute \( x = 7 \) into the expression for \( y \):
\( y = 22 - 3x \)
\( y = 22 - 3(7) \)
\( y = 22 - 21 \)
\( y = 1 \)

Final Solution:
The solution to the system of equations is:
\( x = 7, y = 1 \)

Verification:
Substitute \( x = 7, y = 1 \) into both equations:
1. \( 2x - 7y = 7 \):
\( 2(7) - 7(1) = 14 - 7 = 7 \) (True)
2. \( 3x + y = 22 \):
\( 3(7) + 1 = 21 + 1 = 22 \) (True)

Thus, the solution \( x = 7, y = 1 \) satisfies both equations.