Alright, let's solve this system of linear equations step-by-step.
The system of equations is:
[tex]\[ -6y = 6x + 10 \qquad \text{(1)} \][/tex]
[tex]\[ 5x + 5y = 15 \qquad \text{(2)} \][/tex]
### Step 1: Simplify Equation (2)
First, we'll simplify Equation (2) by dividing all terms by 5:
[tex]\[ x + y = 3 \][/tex]
Thus, we can rewrite Equation (2) as:
[tex]\[ y = 3 - x \qquad \text{(3)} \][/tex]
### Step 2: Substitute Equation (3) into Equation (1)
Now, we substitute [tex]\( y = 3 - x \)[/tex] from Equation (3) into Equation (1):
[tex]\[ -6(3 - x) = 6x + 10 \][/tex]
### Step 3: Solve for [tex]\(x\)[/tex]
Expand and simplify the equation:
[tex]\[ -18 + 6x = 6x + 10 \][/tex]
Next, subtract [tex]\( 6x \)[/tex] from both sides:
[tex]\[ -18 = 10 \][/tex]
This is a contradiction, indicating that the system of equations has no solution. The left-hand side is always [tex]\(-18\)[/tex], while the right-hand side is always [tex]\(10\)[/tex], which means there is no [tex]\(x\)[/tex] or [tex]\(y\)[/tex] that satisfies both equations simultaneously.
### Conclusion
The given system of equations is inconsistent, and there are no values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations. Therefore, the system has no solution.
