Find the perpendicular line of y = 3/2x + 4 that passes through the point (6,2).
Understand the Problem
The question is asking us to find the equation of a line that is perpendicular to the given line (y = 3/2x + 4) and passes through the point (6,2). To solve this, we'll need to determine the slope of the given line first, then find the negative reciprocal to get the slope of the perpendicular line, and finally use the point-slope form to find the equation of the line.
Answer
$$ y = -\frac{2}{3}x + 6 $$
Answer for screen readers
The equation of the line that is perpendicular to $y = \frac{3}{2}x + 4$ and passes through the point (6,2) is:
$$ y = -\frac{2}{3}x + 6 $$
Steps to Solve
- Identify the slope of the given line
The slope of the given line $y = \frac{3}{2}x + 4$ is $\frac{3}{2}$.
- Find the slope of the perpendicular line
To find the slope of a line that is perpendicular to another, take the negative reciprocal of the original slope.
The slope $m$ of the perpendicular line is: $$ m = -\frac{1}{\left(\frac{3}{2}\right)} = -\frac{2}{3} $$
- Use the point-slope form to find the equation of the line
The point-slope form of a line is given by: $$ y - y_1 = m(x - x_1) $$
Here, $(x_1, y_1) = (6, 2)$ and $m = -\frac{2}{3}$. Plugging in these values:
$$ y - 2 = -\frac{2}{3}(x - 6) $$
- Simplify the equation
Distribute $-\frac{2}{3}$ on the right side:
$$ y - 2 = -\frac{2}{3}x + 4 $$
Now, add 2 to both sides to isolate $y$:
$$ y = -\frac{2}{3}x + 4 + 2 $$
This simplifies to:
$$ y = -\frac{2}{3}x + 6 $$
The equation of the line that is perpendicular to $y = \frac{3}{2}x + 4$ and passes through the point (6,2) is:
$$ y = -\frac{2}{3}x + 6 $$
More Information
The slope of the original line is $1.5$, which means it rises 3 units for every 2 units it runs. The perpendicular slope of $-\frac{2}{3}$ indicates that, for every 3 units the line moves to the right, it drops 2 units down.
Tips
- Forgetting to take the negative reciprocal of the slope, which leads to the wrong slope for the perpendicular line. Always ensure to switch the sign and flip the fraction.
- Misapplying the point-slope formula by mixing up the points or signs. Carefully check that you substitute $x_1$ and $y_1$ correctly.
