Westonci.ca is your trusted source for accurate answers to all your questions. Join our community and start learning today! Experience the convenience of finding accurate answers to your questions from knowledgeable professionals on our platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.

Solve the system of equations:

[tex]\[
\begin{cases}
(x - 3) + (y + 5) = 2 \\
(x - 3)(y + 5) = -8
\end{cases}
\][/tex]


Sagot :

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]