We are asked to solve several systems of linear equations. We'll solve system number 6. The system is: $x + 2y = 11$ $2x + 3y = 5$

1. Problem Description

We are asked to solve several systems of linear equations. We'll solve system number

6. The system is:

$x + 2y = 11$

$2x + 3y = 5$

2. Solution Steps

We can use the method of substitution or elimination. Let's use elimination. Multiply the first equation by -2:

$-2(x + 2y) = -2(11)$

$-2x - 4y = -22$

Now we have the following system:

$-2x - 4y = -22$

$2x + 3y = 5$

Add the two equations together:

$(-2x - 4y) + (2x + 3y) = -22 + 5$

$-y = -17$

$y = 17$

Now substitute $y = 17$ into the first original equation:

$x + 2(17) = 11$

$x + 34 = 11$

$x = 11 - 34$

$x = -23$

Therefore, the solution is $x = -23$ and $y = 17$.

We can verify the answer by substituting into the second equation:

$2x + 3y = 5$

$2(-23) + 3(17) = -46 + 51 = 5$

3. Final Answer

$x = -23$, $y = 17$