The problem is to solve the following system of linear equations for $x$ and $y$: $6x + 4y = 7000$ $4x + 8y = 4000$

1. Problem Description

The problem is to solve the following system of linear equations for $x$ and $y$:

$6x + 4y = 7000$

$4x + 8y = 4000$

2. Solution Steps

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

First, we can multiply the first equation by 2 and the second equation by -1 to eliminate $y$:

$2(6x + 4y) = 2(7000)$

$-1(4x + 8y) = -1(4000)$

This gives us:

$12x + 8y = 14000$

$-4x - 8y = -4000$

Now, add the two equations together:

$(12x + 8y) + (-4x - 8y) = 14000 + (-4000)$

$12x - 4x + 8y - 8y = 10000$

$8x = 10000$

Divide by 8:

$x = \frac{10000}{8}$

$x = 1250$

Now that we have the value of $x$, we can substitute it into one of the original equations to find $y$. Let's use the second equation:

$4x + 8y = 4000$

$4(1250) + 8y = 4000$

$5000 + 8y = 4000$

$8y = 4000 - 5000$

$8y = -1000$

$y = \frac{-1000}{8}$

$y = -125$

3. Final Answer

$x = 1250$

$y = -125$