Consider the line $$ y=\frac{5}{4} x+2 $$. Find the equation of the line that is parallel to this line and passes through the point $$ (-4,-4) $$. Find the equation of the line that is perpendicular to this line and passes through the point $$ (-4,-4) $$.

Question

UpStudy AI · Accepted Answer

To solve the problem, we need to find two lines based on the given line $$ y = \frac{5}{4} x + 2 $$:

1. **Find the equation of the line that is parallel to the given line and passes through the point $$ (-4, -4) $$**.

- The slope of the given line is $$ m = \frac{5}{4} $$.
   - Since parallel lines have the same slope, the slope of the new line will also be $$ \frac{5}{4} $$.
   - We can use 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 the line passes through.

Substituting $$ m = \frac{5}{4} $$, $$ x_1 = -4 $$, and $$ y_1 = -4 $$:
   $$
   y - (-4) = \frac{5}{4}(x - (-4))
   $$
   Simplifying this equation will give us the equation of the parallel line.

2. **Find the equation of the line that is perpendicular to the given line and passes through the point $$ (-4, -4) $$**.

- The slope of the perpendicular line is the negative reciprocal of the slope of the given line. Therefore, the slope $$ m' $$ of the perpendicular line is:
     $$
     m' = -\frac{1}{m} = -\frac{1}{\frac{5}{4}} = -\frac{4}{5}
     $$
   - Again, we will use the point-slope form of the equation of a line:
     $$
     y - y_1 = m'(x - x_1)
     $$
   Substituting $$ m' = -\frac{4}{5} $$, $$ x_1 = -4 $$, and $$ y_1 = -4 $$:
   $$
   y - (-4) = -\frac{4}{5}(x - (-4))
   $$
   Simplifying this equation will give us the equation of the perpendicular line.

Now, let's perform the calculations for both lines.

### Parallel Line Calculation
Using the point-slope form:
$$
y + 4 = \frac{5}{4}(x + 4)
$$
Expanding this:
$$
y + 4 = \frac{5}{4}x + 5
$$
Subtracting 4 from both sides:
$$
y = \frac{5}{4}x + 1
$$

### Perpendicular Line Calculation
Using the point-slope form:
$$
y + 4 = -\frac{4}{5}(x + 4)
$$
Expanding this:
$$
y + 4 = -\frac{4}{5}x - \frac{16}{5}
$$
Subtracting 4 (which is $$ \frac{20}{5} $$):
$$
y = -\frac{4}{5}x - \frac{16}{5} - \frac{20}{5}
$$
$$
y = -\frac{4}{5}x - \frac{36}{5}
$$

### Final Results
- The equation of the line parallel to $$ y = \frac{5}{4}x + 2 $$ that passes through $$ (-4, -4) $$ is:
  $$
  y = \frac{5}{4}x + 1
  $$
- The equation of the line perpendicular to $$ y = \frac{5}{4}x + 2 $$ that passes through $$ (-4, -4) $$ is:
  $$
  y = -\frac{4}{5}x - \frac{36}{5}
  $$
The parallel line is $$ y = \frac{5}{4}x + 1 $$ and the perpendicular line is $$ y = -\frac{4}{5}x - \frac{36}{5} $$.

Consider the line \( y=\frac{5}{4} x+2 \). Find the equation of the line that is parallel to this line and passes through the point \( (-4,-4) \). Find the equation of the line that is perpendicular to this line and passes through the point \( (-4,-4) \).

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Algebra Questions