The problem is to solve the following system of linear equations: $2x + 7y = 22$ $5x + 8y = 17$ We need to determine if there is a unique solution, infinitely many solutions, or no solution. If there is a unique solution, we need to find the ordered pair $(x, y)$.

1. Problem Description

The problem is to solve the following system of linear equations:

$2x + 7y = 22$

$5x + 8y = 17$

We need to determine if there is a unique solution, infinitely many solutions, or no solution. If there is a unique solution, we need to find the ordered pair $(x, y)$.

2. Solution Steps

We can use the method of substitution or elimination to solve the system of equations. Let's use elimination.

Multiply the first equation by 5 and the second equation by 2 to eliminate $x$.

$5(2x + 7y) = 5(22)$ => $10x + 35y = 110$

$2(5x + 8y) = 2(17)$ => $10x + 16y = 34$

Subtract the second modified equation from the first modified equation:

$(10x + 35y) - (10x + 16y) = 110 - 34$

$10x + 35y - 10x - 16y = 76$

$19y = 76$

$y = \frac{76}{19} = 4$

Now substitute $y = 4$ into the first equation $2x + 7y = 22$:

$2x + 7(4) = 22$

$2x + 28 = 22$

$2x = 22 - 28$

$2x = -6$

$x = \frac{-6}{2} = -3$

So, the solution is $x = -3$ and $y = 4$. The ordered pair is $(-3, 4)$.

We can check the solution in the second equation:

$5x + 8y = 17$

$5(-3) + 8(4) = -15 + 32 = 17$. This is correct.

3. Final Answer

The solution to the system is $(-3, 4)$.