Certainly! Let's solve the given system of equations to find the solutions for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. We will solve the following equations step-by-step:
1. [tex]\( x - 7 = \frac{2}{3}(y + 15) \)[/tex]
2. [tex]\( y = \frac{2}{3} x + 17 \)[/tex]
3. [tex]\( y - 7 = \frac{2}{3}(x + 15) \)[/tex]
4. [tex]\( 6x - 3y = -51 \)[/tex]
5. [tex]\( -\frac{2}{3} x + y = 17 \)[/tex]
### Step 1: Simplify the Equations
First, let's simplify each equation into a standard form [tex]\( ax + by = c \)[/tex].
1. [tex]\( x - 7 = \frac{2}{3}(y + 15) \)[/tex]
[tex]\[ x - 7 = \frac{2}{3}y + 10 \][/tex]
[tex]\[ \Rightarrow x - \frac{2}{3}y = 17 \quad \text{(after moving terms around)} \][/tex]
2. [tex]\( y = \frac{2}{3} x + 17 \)[/tex]
[tex]\[ y = \frac{2}{3}x + 17 \][/tex]
[tex]\[ \Rightarrow \frac{2}{3}x - y = -17 \][/tex]
3. [tex]\( y - 7 = \frac{2}{3}(x + 15) \)[/tex]
[tex]\[ y - 7 = \frac{2}{3}x + 10 \][/tex]
[tex]\[ y = \frac{2}{3}x + 17 \][/tex]
[tex]\[ \Rightarrow \frac{2}{3}x - y = -17 \][/tex]
4. [tex]\( 6x - 3y = -51 \)[/tex]
5. [tex]\( -\frac{2}{3} x + y = 17 \)[/tex]
### Step 2: Solve the Simplified System
Now we will solve the simplified system of equations. Notice the cases where destructive interference might help:
1. [tex]\( x - \frac{2}{3}y = 17 \)[/tex]
2. [tex]\( \frac{2}{3}x - y = -17 \)[/tex]
3. [tex]\( \frac{2}{3}x - y = -17 \)[/tex] (same as equation 2)
4. [tex]\( 6x - 3y = -51 \)[/tex]
5. [tex]\( -\frac{2}{3}x + y = 17 \)[/tex] (same as equation 2 but rearranged)
### Step 3: Identify Redundancies and Conflicts
Let's analyze if there are conflicts or redundancies in equations.
- Equation (2) and (3) are identical.
- Equation (5) is just equation (2) rephrased.
Now we can group the unique equations:
1. [tex]\( x - \frac{2}{3}y = 17 \)[/tex]
2. [tex]\( \frac{2}{3}x - y = -17 \)[/tex]
3. [tex]\( 6x - 3y = -51 \)[/tex]
### Step 4: Consistency Check
Let's check the consistency by considering these 3:
1. [tex]\( x - \frac{2}{3} y = 17 \)[/tex]
2. [tex]\( \frac{2}{3} x - y = -17 \)[/tex]
Let's multiply equation (2) by 3 to align coefficients:
[tex]\[ 2x - 3y = -51 \][/tex]
Now, we compare with equation (4):
[tex]\[ 6x - 3y = -51 \][/tex]
Divide the above by 3:
[tex]\[ 2x - y = -17 \][/tex]
This result compares with the previous derivation directly rules out all real numbers fulfilling these while maintaining consistency for all given conditions, indicating the computed result reflected all and having redundancy causing undefined intersection points across consistent equations.
### Conclusion
After examining the entire system of linear equations given, we determine there are no solutions [tex]\( x \)[/tex] and [tex]\( y \)[/tex] where all equations can be made true. This means the system of equations is inconsistent. No (x,y) pair satisfies all of them simultaneously.
Therefore, the final result is:
[tex]\[
\boxed{[]}
\][/tex]
or there are no solutions as indicated.
