Find the parallel line of y = 1/3x + 2 that passes through the point (-3, 4).
Understand the Problem
The question is asking us to find the equation of a line that is parallel to the given line, y = (1/3)x + 2, and passes through the specified point (-3, 4). Since parallel lines have the same slope, we will use the slope from the given line and apply the point-slope formula to determine the new line's equation.
Answer
$y = \frac{1}{3}x + 5$
Answer for screen readers
The equation of the line parallel to $y = \frac{1}{3}x + 2$ and passing through the point (-3, 4) is $y = \frac{1}{3}x + 5$.
Steps to Solve
- Identify the slope of the given line
The given line is $y = \frac{1}{3}x + 2$. The slope (m) of this line is $\frac{1}{3}$.
- Use the point-slope formula
The point-slope form of a line is given by the equation:
$$ y - y_1 = m(x - x_1) $$
where $(x_1, y_1)$ is a point on the line and $m$ is the slope. We know $(x_1, y_1) = (-3, 4)$ and $m = \frac{1}{3}$.
- Substitute the known values into the formula
Substituting $m$, $x_1$, and $y_1$ into the point-slope formula, we get:
$$ y - 4 = \frac{1}{3}(x + 3) $$
- Simplify the equation to slope-intercept form
Distributing the slope on the right side and solving for $y$ gives:
$$ y - 4 = \frac{1}{3}x + 1 $$
Now, adding 4 to both sides:
$$ y = \frac{1}{3}x + 5 $$
This is the equation of the new line.
The equation of the line parallel to $y = \frac{1}{3}x + 2$ and passing through the point (-3, 4) is $y = \frac{1}{3}x + 5$.
More Information
This tells us that the new line has the same slope as the original line, ensuring they remain parallel. The y-intercept of the new line is 5, indicating where it crosses the y-axis.
Tips
- Forgetting to use the correct slope: Always remember that parallel lines have the same slope as the original line.
- Misapplying the point-slope formula: Make sure to substitute the coordinates correctly into the formula.
