Looking for answers? Westonci.ca is your go-to Q&A platform, offering quick, trustworthy responses from a community of experts. Connect with a community of experts ready to provide precise solutions to your questions quickly and accurately. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
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]
### 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]
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.
