You have a system of linear equations:

1. $8x + 3y = 13$
2. $3x + 2y = 11$

To solve this, we can use either substitution or elimination. Let’s use the elimination method to solve for $x$ and $y$.

Step 1: Eliminate one of the variables

We will multiply the second equation by 4 and the first equation by 3, to make the coefficients of $y$ the same:

• Multiply the second equation $3x + 2y = 11$ by 4: $4(3x + 2y) = 4(11) \implies 12x + 8y = 44$

• Multiply the first equation $8x + 3y = 13$ by 3: $3(8x + 3y) = 3(13) \implies 24x + 9y = 39$

Now, the system of equations is:

1. $24x + 9y = 39$
2. $12x + 8y = 44$

Step 2: Subtract the equations

Next, we subtract the second equation from the first: $(24x + 9y) - (12x + 8y) = 39 - 44$ $12x + y = -5$ Thus, we have the new equation: $12x + y = -5 \quad \text{(Equation 3)}$

Step 3: Solve for $y$

From equation 3: $y = -5 - 12x$

Step 4: Substitute $y$ into one of the original equations

Substitute $y = -5 - 12x$ into the second original equation $3x + 2y = 11$:

$3x + 2(-5 - 12x) = 11$ $3x - 10 - 24x = 11$ $-21x = 21$ $x = -1$

Step 5: Solve for $y$

Now substitute $x = -1$ into $y = -5 - 12x$:

$y = -5 - 12(-1)$ $y = -5 + 12$ $y = 7$

Final Solution

The solution to the system of equations is: $x = -1, \quad y = 7$

Would you like further details or clarification on any of the steps? Here's a list of related questions:

1. What is the substitution method and when should it be used?
2. Can you use matrices to solve a system of linear equations?
3. What would happen if the system had no solution?
4. How can you check your solution for correctness?
5. What is the graphical interpretation of solving a system of linear equations?

Tip: Always try to simplify one of the equations before starting the elimination method. It can make the process faster!