To determine whether the given system of equations is consistent and dependent or independent, we need to analyze the system step-by-step through algebraic manipulation.
Given the system of equations is:
[tex]\[ y = -\frac{1}{3}x + 2 \][/tex]
[tex]\[ x + 3y = 4 \][/tex]
### Step 1: Substitute the expression for [tex]\( y \)[/tex] from the first equation into the second equation.
From the first equation:
[tex]\[ y = -\frac{1}{3}x + 2 \][/tex]
Substitute this into the second equation:
[tex]\[ x + 3\left(-\frac{1}{3}x + 2\right) = 4 \][/tex]
### Step 2: Simplify the substituted equation.
Simplify the left side:
[tex]\[ x + 3(-\frac{1}{3}x + 2) = 4 \][/tex]
[tex]\[ x + (-x + 6) = 4 \][/tex]
[tex]\[ x - x + 6 = 4 \][/tex]
[tex]\[ 6 = 4 \][/tex]
### Step 3: Analyze the simplified equation.
We end up with the statement:
[tex]\[ 6 = 4 \][/tex]
This is a contradiction, meaning this equation is never true. The resulting contradiction indicates that there is no solution that satisfies both equations simultaneously.
### Conclusion
Since the system of equations leads to a contradiction, it is:
- Inconsistent: There is no solution that satisfies both equations.
- Independent/Dependent: This classification only applies to consistent systems. Since our system is inconsistent, this classification isn't applicable here.
Therefore, the correct classification of this system of equations is:
Inconsistent only
