Answered

Welcome to Westonci.ca, where you can find answers to all your questions from a community of experienced professionals. Explore our Q&A platform to find reliable answers from a wide range of experts in different fields. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.

Consider the line [tex]$y=\frac{2}{7}x-6$[/tex].

1. Find the equation of the line that is parallel to this line and passes through the point [tex]$(4,3)$[/tex].
2. Find the equation of the line that is perpendicular to this line and passes through the point [tex]$(4,3)$[/tex].

Equation of parallel line: [tex]$\square$[/tex]

Equation of perpendicular line: [tex]$\square$[/tex]

Sagot :

GinnyAnswer
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]
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.