Find the perpendicular line of y = -2x + 7 that passes through the point (6, 2).
Understand the Problem
The question is asking to find the equation of a line that is perpendicular to the given line y = -2x + 7 and passes through the specified point (6,2). To do this, we need to determine the slope of the given line, then find the negative reciprocal of that slope to get the slope of the perpendicular line. Finally, we can use the point-slope form to write the equation of the perpendicular line.
Answer
The equation of the perpendicular line is $$ y = \frac{1}{2}x - 1 $$.
Answer for screen readers
The equation of the line that is perpendicular to $y = -2x + 7$ and passes through the point (6,2) is:
$$ y = \frac{1}{2}x - 1 $$
Steps to Solve
- Find the slope of the given line
The given line is $y = -2x + 7$. The slope (m) of this line is -2.
- Determine the slope of the perpendicular line
To find the slope of a line that is perpendicular, we take the negative reciprocal of the original slope. The negative reciprocal of -2 is:
$$ m_\perpendicular = -\frac{1}{-2} = \frac{1}{2} $$
- Use the point-slope form of the equation of a line
Now that we have the slope of the perpendicular line ($m_\perpendicular = \frac{1}{2}$) and a point that it passes through (6,2), we can use the point-slope form:
$$ y - y_1 = m(x - x_1) $$
Plugging in the values $(x_1, y_1) = (6, 2)$:
$$ y - 2 = \frac{1}{2}(x - 6) $$
- Rearrange to find the equation in slope-intercept form
Next, we can simplify this equation to get it into slope-intercept form ($y = mx + b$):
$$ y - 2 = \frac{1}{2}x - 3 $$
Adding 2 to both sides gives:
$$ y = \frac{1}{2}x - 1 $$
The equation of the line that is perpendicular to $y = -2x + 7$ and passes through the point (6,2) is:
$$ y = \frac{1}{2}x - 1 $$
More Information
This equation represents a line with a slope of $\frac{1}{2}$ that intersects the y-axis at -1. Perpendicular lines have slopes that multiply to -1, confirming that this line is indeed perpendicular to the original.
Tips
- Confusing slope calculations: Ensure that the correct negative reciprocal is used.
- Misapplying the point-slope form: Remember to substitute the correct point into the equation.
- Forgetting to simplify the equation: It’s important to rearrange the final equation into slope-intercept form.
