Solve the following system of equations for "x". Express your answer as a decimal number with **at least four digits past the decimal.** 

y - (3.9)(x) = (4.3)(x + y) + 4.1

y + 8.8 = (y - x)/2

We are given a system of two equations:

1. \( y - 3.9x = 4.3(x + y) + 4.1 \)
2. \( y + 8.8 = \frac{y - x}{2} \)

Let's solve this step by step.

### Step 1: Simplify the first equation
From the first equation:

\[
y - 3.9x = 4.3(x + y) + 4.1
\]
Expand the right-hand side:
\[
y - 3.9x = 4.3x + 4.3y + 4.1
\]
Now, rearrange the equation by bringing all terms involving \(x\) and \(y\) to the left-hand side:
\[
y - 3.9x - 4.3x - 4.3y = 4.1
\]
Simplify:
\[
-3.3y - 8.2x = 4.1
\]
This simplifies to:
\[
3.3y + 8.2x = -4.1
\]
Thus, the first equation is:
\[
3.3y + 8.2x = -4.1 \quad \text{(Equation 1)}
\]

### Step 2: Simplify the second equation
From the second equation:
\[
y + 8.8 = \frac{y - x}{2}
\]
Multiply both sides by 2 to eliminate the fraction:
\[
2(y + 8.8) = y - x
\]
Expand the left-hand side:
\[
2y + 17.6 = y - x
\]
Now, move terms involving \(y\) and \(x\) to the same side:
\[
2y - y + x = -17.6
\]
Simplify:
\[
y + x = -17.6
\]
Thus, the second equation is:
\[
y + x = -17.6 \quad \text{(Equation 2)}
\]

### Step 3: Solve the system of equations
We now have the system:
1. \( 3.3y + 8.2x = -4.1 \)
2. \( y + x = -17.6 \)

#### Substitution method:
From Equation 2, solve for \(y\):
\[
y = -17.6 - x
\]
Substitute this expression for \(y\) into Equation 1:
\[
3.3(-17.6 - x) + 8.2x = -4.1
\]
Expand:
\[
3.3(-17.6) - 3.3x + 8.2x = -4.1
\]
Calculate \(3.3 \times -17.6\):
\[
-58.08 - 3.3x + 8.2x = -4.1
\]
Combine like terms:
\[
-58.08 + 4.9x = -4.1
\]
Now, isolate \(x\):
\[
4.9x = -4.1 + 58.08
\]
\[
4.9x = 53.98
\]
\[
x = \frac{53.98}{4.9}
\]
\[
x \approx 11.0143
\]

### Step 4: Find \(y\)
Substitute \(x = 11.0143\) into Equation 2:
\[
y + 11.0143 = -17.6
\]
\[
y = -17.6 - 11.0143
\]
\[
y \approx -28.6143
\]

### Final Answer:
The solution is approximately:
\[
x \approx 11.0143, \quad y \approx -28.6143
\]

Would you like more details or have any questions? 

Here are some related questions to consider:
1. How do you solve linear systems using the substitution method?
2. What is the elimination method for solving systems of equations?
3. How can graphing help in solving systems of equations?
4. What are some applications of solving systems of linear equations?
5. How do you interpret the solution of a system of equations geometrically?

**Tip:** Double-check your equations for transcription errors before solving, as small mistakes can lead to large discrepancies in the final answer.

Solve the following system of equations for 'x'. Express your answer as a decimal number with at least four digits past the decimal.

y - (3.9)(x) = (4.3)(x + y) + 4.1

y + 8.8 = (y - x)/2

Learn how to solve a system of linear equations using substitution and simplification methods. This example involves expressing the answer as a decimal to at least four digits past the decimal. Suitable for students in Grades 9-12.

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