Question

Write a​ slope-intercept equation for a line passing through the point ​(44​, 1919​) that is parallel to yequals=one half
1
2xplus+1111. Then write a second equation for a line passing through the given point that is perpendicular to the given line.
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Part 1
Which answer below is​ correct?
A.parallel: y equals one half x plus 17 perpendicular: y equals negative 2 x plus 27
parallel: y equals one half x plus 17 perpendicular: y equals negative 2 x plus 27parallel: y=
1
2x+17 perpendicular: y=−2x+27
B.parallel: y equals one half x plus 19 perpendicular: y equals negative 2 x plus 19
parallel: y equals one half x plus 19 perpendicular: y equals negative 2 x plus 19parallel: y=
1
2x+19 perpendicular: y=−2x+19
C.parallel: y equals one half x plus 17 perpendicular: y equals 2 x plus 27
parallel: y equals one half x plus 17 perpendicular: y equals 2 x plus 27parallel: y=
1
2x+17 perpendicular: y=2x+27

Answer 1

To find the equations for the lines that are parallel and perpendicular to the given line passing through the point \((44, 1919)\), we need to follow these steps:

1. **Identify the slope of the original line**:
The given line is in the slope-intercept form \(y = \frac{1}{2}x + 1111\). The slope (\(m\)) of this line is \(\frac{1}{2}\).

2. **Find the equation of the parallel line**:
Lines that are parallel have the same slope. Therefore, the parallel line will also have a slope of \(\frac{1}{2}\). We can use point-slope form to write the equation:
\[
y - y_1 = m(x - x_1)
\]
Plugging in the point \((44, 1919)\) and the slope \(\frac{1}{2}\):
\[
y - 1919 = \frac{1}{2}(x - 44)
\]
Distributing:
\[
y - 1919 = \frac{1}{2}x - 22
\]
Adding \(1919\) to both sides gives:
\[
y = \frac{1}{2}x + 1897
\]

3. **Find the equation of the perpendicular line**:
The slope of a line that is perpendicular to another line is the negative reciprocal. The slope of the original line is \(\frac{1}{2}\), so the slope of the perpendicular line is \(-2\) (the negative reciprocal of \(\frac{1}{2}\)). Again, using the point-slope form:
\[
y - 1919 = -2(x - 44)
\]
Distributing:
\[
y - 1919 = -2x + 88
\]
Adding \(1919\) to both sides gives:
\[
y = -2x + 2007
\]

So, the equations of the lines are:
- Parallel: \(y = \frac{1}{2}x + 1897\)
- Perpendicular: \(y = -2x + 2007\)

None of the options given (A, B, or C) matches these results. If you need to adjust values in the options or double-check the given equations, please check again.