We are given the following system of linear equations:

\[
384x + 240y = 1632
\]
\[
480x + 640y = 2720
\]

We can solve this system using either substitution, elimination, or matrix methods. I'll use **elimination** here to simplify the process.

### Step 1: Simplify the equations (if possible)
First, let's see if we can reduce the equations by dividing by any common factors.

- In the first equation, \( 384x + 240y = 1632 \), all terms are divisible by 48:
  \[
  \frac{384x + 240y}{48} = \frac{1632}{48}
  \]
  This simplifies to:
  \[
  8x + 5y = 34
  \]

- In the second equation, \( 480x + 640y = 2720 \), all terms are divisible by 160:
  \[
  \frac{480x + 640y}{160} = \frac{2720}{160}
  \]
  This simplifies to:
  \[
  3x + 4y = 17
  \]

So, the simplified system of equations is:

\[
8x + 5y = 34 \tag{1}
\]
\[
3x + 4y = 17 \tag{2}
\]

### Step 2: Elimination Method
We will eliminate one of the variables by making the coefficients of either \(x\) or \(y\) equal in both equations. Let's eliminate \(x\).

- Multiply equation (2) by 8:
  \[
  8(3x + 4y = 17) \implies 24x + 32y = 136 \tag{3}
  \]

- Multiply equation (1) by 3:
  \[
  3(8x + 5y = 34) \implies 24x + 15y = 102 \tag{4}
  \]

Now subtract equation (4) from equation (3):
\[
(24x + 32y) - (24x + 15y) = 136 - 102
\]
This simplifies to:
\[
17y = 34
\]
Solving for \(y\):
\[
y = \frac{34}{17} = 2
\]

### Step 3: Solve for \(x\)
Substitute \(y = 2\) back into one of the original simplified equations, say \(3x + 4y = 17\):
\[
3x + 4(2) = 17
\]
\[
3x + 8 = 17
\]
\[
3x = 17 - 8
\]
\[
3x = 9
\]
\[
x = \frac{9}{3} = 3
\]

### Final Solution:
The solution to the system is:
\[
x = 3, \quad y = 2
\]

Let me know if you would like further details or have any questions!

Here are 5 related questions to deepen your understanding:
1. How do you solve a system of equations using the substitution method?
2. What is the advantage of using elimination over substitution in solving systems?
3. How can you represent systems of equations graphically?
4. What is the determinant, and how does it relate to solving linear systems?
5. Can systems of equations have no solution or infinitely many solutions? When does that happen?

**Tip:** Always check if you can simplify the equations first—it makes solving much easier!

Discover how to solve the system of linear equations 384x + 240y = 1632 and 480x + 640y = 2720 using the elimination method. The solution simplifies the equations and demonstrates step-by-step elimination to find x = 3 and y = 2. Suitable for students in Grades 9-11.

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