Let's analyze the problem step by step.
### Step 1: Equation of the given line
The given line equation is:
[tex]\[ y = \frac{2}{7}x - 6 \][/tex]
The slope of this line is \(\frac{2}{7}\).
### Step 2: Parallel Line through (4, 3)
For a line to be parallel to the given line, it must have the same slope. So, the slope of the parallel line will also be \(\frac{2}{7}\).
Using the point-slope form of the line equation:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where \((x_1, y_1)\) is the point the line passes through and \(m\) is the slope. Here, \((x_1, y_1) = (4, 3)\) and \(m = \frac{2}{7}\).
Substitute the values:
[tex]\[ y - 3 = \frac{2}{7}(x - 4) \][/tex]
Solving for \(y\):
[tex]\[ y - 3 = \frac{2}{7}x - \frac{8}{7} \][/tex]
[tex]\[ y = \frac{2}{7}x - \frac{8}{7} + 3 \][/tex]
[tex]\[ y = \frac{2}{7}x - \frac{8}{7} + \frac{21}{7} \][/tex]
[tex]\[ y = \frac{2}{7}x + \frac{13}{7} \][/tex]
So, the equation of the parallel line is:
[tex]\[ y = 0.285714285714286 x + 1.85714285714286 \][/tex]
### Step 3: Perpendicular Line through (4, 3)
For a line to be perpendicular to the given line, its slope must be the negative reciprocal of the slope of the given line. The slope of the given line is \(\frac{2}{7}\), so the slope of the perpendicular line will be:
[tex]\[ m_{\perpendicular} = -\frac{7}{2} \][/tex]
Using the same point-slope form:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Substitute \((x_1, y_1) = (4, 3)\) and \(m = -\frac{7}{2}\):
[tex]\[ y - 3 = -\frac{7}{2}(x - 4) \][/tex]
Solving for \(y\):
[tex]\[ y - 3 = -\frac{7}{2}x + 14 \][/tex]
[tex]\[ y = -\frac{7}{2}x + 14 + 3 \][/tex]
[tex]\[ y = -\frac{7}{2}x + 17 \][/tex]
So, the equation of the perpendicular line is:
[tex]\[ y = 17 - 3.5 x \][/tex]
### Summary of Results
- Equation of the parallel line: [tex]\[ y = 0.285714285714286 x + 1.85714285714286 \][/tex]
- Equation of the perpendicular line: [tex]\[ y = 17 - 3.5 x \][/tex]
Feel free to fill these results into the squares provided in your question:
Equation of parallel line:
[tex]\[
y = 0.285714285714286 x + 1.85714285714286
\][/tex]
Equation of perpendicular line:
[tex]\[
y = 17 - 3.5 x
\][/tex]
