To solve the system of equations using the elimination method, we start with the given system:

-3x + 8y = -35 \tag{1} -2x + 3y = -14 \tag{2}

Step 1: Eliminate one of the variables

We need to manipulate the equations in such a way that when we add or subtract them, one of the variables cancels out. Let's eliminate $x$.

To do this, we need to find a common coefficient for $x$ in both equations. The least common multiple (LCM) of 3 and 2 is 6. So, we will multiply equation (1) by 2 and equation (2) by 3 to make the coefficients of $x$ the same:

Multiply equation (1) by 2: $2(-3x + 8y) = 2(-35)$ -6x + 16y = -70 \tag{3}

Multiply equation (2) by 3: $3(-2x + 3y) = 3(-14)$ -6x + 9y = -42 \tag{4}

Step 2: Subtract the two equations

Now, subtract equation (4) from equation (3) to eliminate $x$:

$(-6x + 16y) - (-6x + 9y) = -70 - (-42)$ $-6x + 16y + 6x - 9y = -70 + 42$ $7y = -28$

Step 3: Solve for $y$

Now solve for $y$:

$y = \frac{-28}{7} = -4$

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

Now that we know $y = -4$, we can substitute this value into either of the original equations to solve for $x$. We'll use equation (2):

$-2x + 3y = -14$ Substitute $y = -4$: $-2x + 3(-4) = -14$ $-2x - 12 = -14$ $-2x = -14 + 12$ $-2x = -2$ $x = \frac{-2}{-2} = 1$

Final Solution:

The solution to the system of equations is $x = 1$ and $y = -4$, so the ordered pair is:

$(1, -4)$

Summary:

The ordered pair that solves the system of equations is $(1, -4)$.

Would you like more details or further explanations on any part of the process? Here are some related questions for you:

1. What would happen if we used substitution instead of elimination to solve this system?
2. How do you determine when to use elimination versus substitution in solving a system of equations?
3. How can you verify that the solution $(1, -4)$ is correct for both equations?
4. What would the solution look like if the system had no solution or infinitely many solutions?
5. How do you solve a system of equations where both equations represent parallel lines?

Tip: When using elimination, always look for the smallest common multiple of the coefficients of one variable to simplify the process.