To solve the given system of equations:
[tex]\[
\left\{
\begin{array}{l}
(x - 3) + (y + 5) = 2 \\
(x - 3)(y + 5) = -8
\end{array}
\right.
\][/tex]
Let's break it down step by step.
### Step 1: Simplify the First Equation
First, we simplify the first equation:
[tex]\[
(x - 3) + (y + 5) = 2
\][/tex]
Combine like terms:
[tex]\[
x - 3 + y + 5 = 2 \\
x + y + 2 = 2
\][/tex]
Subtract 2 from both sides to isolate [tex]\(x + y\)[/tex]:
[tex]\[
x + y = 0
\][/tex]
### Step 2: Express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]
From [tex]\(x + y = 0\)[/tex], we can solve for [tex]\(y\)[/tex]:
[tex]\[
y = -x
\][/tex]
### Step 3: Substitute [tex]\( y = -x \)[/tex] into the Second Equation
Next, we substitute [tex]\(y = -x\)[/tex] into the second equation:
[tex]\[
(x - 3)(y + 5) = -8
\][/tex]
Substitute [tex]\(y = -x\)[/tex]:
[tex]\[
(x - 3)(-x + 5) = -8
\][/tex]
Distribute to simplify:
[tex]\[
(x - 3)(5 - x) = -8
\][/tex]
Expand the expression:
[tex]\[
5x - x^2 - 15 + 3x = -8
\][/tex]
Combine like terms:
[tex]\[
-x^2 + 8x - 15 = -8
\][/tex]
Move everything to one side to form a quadratic equation:
[tex]\[
-x^2 + 8x - 15 + 8 = 0 \\
-x^2 + 8x - 7 = 0
\][/tex]
Multiply through by -1 to simplify:
[tex]\[
x^2 - 8x + 7 = 0
\][/tex]
### Step 4: Solve the Quadratic Equation
To solve the quadratic equation [tex]\(x^2 - 8x + 7 = 0\)[/tex], we can factorize it:
[tex]\[
x^2 - 8x + 7 = (x - 1)(x - 7) = 0
\][/tex]
So, setting each factor equal to zero gives:
[tex]\[
x - 1 = 0 \quad \text{or} \quad x - 7 = 0
\][/tex]
Therefore, the solutions for [tex]\(x\)[/tex] are:
[tex]\[
x = 1 \quad \text{or} \quad x = 7
\][/tex]
### Step 5: Find Corresponding [tex]\( y \)[/tex] Values
Now, we find the corresponding [tex]\(y\)[/tex] values using [tex]\(y = -x\)[/tex]:
- If [tex]\(x = 1\)[/tex]:
[tex]\[
y = -1
\][/tex]
- If [tex]\(x = 7\)[/tex]:
[tex]\[
y = -7
\][/tex]
### Final Solutions
Therefore, the two solutions for the system of equations are:
[tex]\[
(x, y) = (1, -1) \quad \text{and} \quad (7, -7)
\][/tex]
So, the solutions to the given system of equations are:
[tex]\[
(1, -1) \quad \text{and} \quad (7, -7)
\][/tex]
