Question
Find equations of the lines that pass through the given point and are parallel to and perpendicular to the given line. \( 8 x-2 y=5, \quad(4,1) \) (a) parallel to the given line (b) perpendicular to the given line (a)
Ask by Turnbull Brooks. in the United States
Feb 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The equations are:
- Parallel: \( y = 4x - 15 \)
- Perpendicular: \( y = -\frac{1}{4}x + 2 \)
Solution
To find the equations of the lines that pass through the point \((4, 1)\) and are parallel to and perpendicular to the given line \(8x - 2y = 5\), we will follow these steps:
### Step 1: Find the slope of the given line
First, we need to rewrite the equation of the given line in slope-intercept form \(y = mx + b\), where \(m\) is the slope.
Starting with the equation:
\[
8x - 2y = 5
\]
We can solve for \(y\):
\[
-2y = -8x + 5
\]
\[
y = 4x - \frac{5}{2}
\]
From this, we can see that the slope \(m\) of the given line is \(4\).
### Step 2: Find the equation of the line parallel to the given line
Lines that are parallel have the same slope. Therefore, the slope of the line we want to find is also \(4\).
Using the point-slope form of the equation of a line, which is given by:
\[
y - y_1 = m(x - x_1)
\]
where \((x_1, y_1)\) is the point through which the line passes, we can substitute \(m = 4\) and the point \((4, 1)\):
\[
y - 1 = 4(x - 4)
\]
Now, we can simplify this equation:
\[
y - 1 = 4x - 16
\]
\[
y = 4x - 15
\]
Thus, the equation of the line that is parallel to the given line and passes through the point \((4, 1)\) is:
\[
\boxed{y = 4x - 15}
\]
### Step 3: Find the equation of the line perpendicular to the given line
Lines that are perpendicular have slopes that are negative reciprocals of each other. The negative reciprocal of \(4\) is \(-\frac{1}{4}\).
Using the point-slope form again with the slope \(-\frac{1}{4}\):
\[
y - 1 = -\frac{1}{4}(x - 4)
\]
Now, we can simplify this equation:
\[
y - 1 = -\frac{1}{4}x + 1
\]
\[
y = -\frac{1}{4}x + 2
\]
Thus, the equation of the line that is perpendicular to the given line and passes through the point \((4, 1)\) is:
\[
\boxed{y = -\frac{1}{4}x + 2}
\]
In summary:
- The equation of the line parallel to the given line is \(y = 4x - 15\).
- The equation of the line perpendicular to the given line is \(y = -\frac{1}{4}x + 2\).
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Beyond the Answer
To find the equation of the line parallel to \(8x - 2y = 5\) that passes through the point \((4, 1)\), we first need to determine the slope of the given line. Rearranging to slope-intercept form \(y = mx + b\): \[ -2y = -8x + 5 \] \[ y = 4x - \frac{5}{2} \] The slope \(m\) is \(4\). Since parallel lines have the same slope, the equation of the line is: \[ y - 1 = 4(x - 4) \] Simplifying gives: \[ y = 4x - 15 \] (b) Now, for the line perpendicular to the given line, we take the negative reciprocal of the slope \(4\), which is \(-\frac{1}{4}\). Using the point-slope form through point \((4, 1)\): \[ y - 1 = -\frac{1}{4}(x - 4) \] Solving this gives: \[ y = -\frac{1}{4}x + 2 \]
