Find the perpendicular line of y = -2/3x - 3 that passes through the point (4, 8).
Understand the Problem
The question is asking us to find the equation of a line that is perpendicular to the given line y = -2/3x - 3 and passes through the point (4, 8). To approach this, we need to determine the slope of the given line, then find the negative reciprocal of that slope for the perpendicular line, and use the point-slope form to write the equation of the new line.
Answer
The equation of the line is $$ y = \frac{3}{2}x + 2 $$
Answer for screen readers
The equation of the line that is perpendicular to the given line and passes through the point (4, 8) is
$$ y = \frac{3}{2}x + 2 $$
Steps to Solve
- Find the slope of the given line
The given line is $y = -\frac{2}{3}x - 3$. The slope of this line is $m = -\frac{2}{3}$.
- Determine the slope of the perpendicular line
To find the slope of the line that is perpendicular to this line, we need to take the negative reciprocal of $-\frac{2}{3}$.
The negative reciprocal is calculated as follows:
$$ m_{perpendicular} = -\frac{1}{-\frac{2}{3}} = \frac{3}{2} $$
- Use the point-slope form for the new line
We will use the point-slope formula, which is given by:
$$ y - y_1 = m(x - x_1) $$
Here, $(x_1, y_1) = (4, 8)$ and $m_{perpendicular} = \frac{3}{2}$. Substituting these values into the point-slope formula gives:
$$ y - 8 = \frac{3}{2}(x - 4) $$
- Simplify the equation
Now, we simplify the equation:
First, distribute $\frac{3}{2}$:
$$ y - 8 = \frac{3}{2}x - \frac{3}{2} \cdot 4 $$
Calculating the second term gives:
$$ y - 8 = \frac{3}{2}x - 6 $$
Finally, add 8 to both sides:
$$ y = \frac{3}{2}x - 6 + 8 $$
This simplifies to:
$$ y = \frac{3}{2}x + 2 $$
The equation of the line that is perpendicular to the given line and passes through the point (4, 8) is
$$ y = \frac{3}{2}x + 2 $$
More Information
This line has a slope of $\frac{3}{2}$ and intersects the y-axis at 2. Perpendicular lines have slopes that multiply to -1, and here, $-\frac{2}{3}$ and $\frac{3}{2}$ maintain that relationship.
Tips
- Confusing the slope of the perpendicular line by not taking the negative reciprocal.
- Misplacing the point in the point-slope formula. Always check that you substitute the correct coordinates of the given point (4, 8) in the formula.
