Solve this system of equations using elimination method.Must show work/steps on your work page {6𝑥 +6𝑦=−65𝑥 + 𝑦 =−13(, )

1. Break Down the Problem

We have the following system of equations to solve using the elimination method:

1. $6x + 6y = -65$
2. $-5x + y = -13$

2. Relevant Concepts

The elimination method involves manipulating the equations to eliminate one variable, allowing us to solve for the other variable.

3. Analysis and Detail

Let's rearrange and manipulate the equations step by step.

3.1 Rewrite the Equations

1. From the first equation: $6x + 6y = -65 \quad (1)$
2. From the second equation: $-5x + y = -13 \quad (2)$

3.2 Eliminate $y$

To eliminate $y$, we can manipulate both equations. We can make the coefficients of $y$ equal by multiplying equation (2) by 6:

$6(-5x + y) = 6(-13)$ This gives us: $-30x + 6y = -78 \quad (3)$

3.3 Set Up for Elimination

We now have:

1. $6x + 6y = -65$
2. $-30x + 6y = -78$

Next, we can subtract equation (1) from equation (3):

$(-30x + 6y) - (6x + 6y) = -78 - (-65)$

3.4 Simplify

$-30x + 6y - 6x - 6y = -78 + 65$ This simplifies to: $-36x = -13$

3.5 Solve for $x$

$x = \frac{-13}{-36} = \frac{13}{36}$

4. Verify and Summarize

Now, substitute $x = \frac{13}{36}$ back into either original equation to solve for $y$. Let's use equation (2).

4.1 Substitute $x$ into Equation (2)

$-5\left(\frac{13}{36}\right) + y = -13$ This gives us: $-\frac{65}{36} + y = -13$

4.2 Solve for $y$

$y = -13 + \frac{65}{36}$ Convert $-13$ to a fraction: $y = -\frac{468}{36} + \frac{65}{36}$ $y = \frac{65 - 468}{36} = \frac{-403}{36}$

Final Answer

The solution to the system of equations is: $x = \frac{13}{36}, \quad y = \frac{-403}{36}$