Find the parallel line of y=1/2x that passes through the point (2, 4).
Understand the Problem
The question is asking to find a line that is parallel to the given line y=1/2x and also passes through the point (2, 4). The approach involves using the slope of the given line and applying the point-slope form of a line equation.
Answer
The equation of the line is $y = \frac{1}{2}x + 3$.
Answer for screen readers
The equation of the line parallel to $y = \frac{1}{2}x$ that passes through the point (2, 4) is $y = \frac{1}{2}x + 3$.
Steps to Solve
- Identify the slope of the given line
The given line equation is $y = \frac{1}{2}x$. The slope (m) is the coefficient of $x$, which is $\frac{1}{2}$.
- Use the point-slope formula
To find the equation of a line that is parallel to the given line and passes through the point (2, 4), we use the point-slope form of the line equation, which is given by: $$ y - y_1 = m(x - x_1) $$ Here, $(x_1, y_1) = (2, 4)$ and $m = \frac{1}{2}$.
- Substitute the values into the point-slope formula
Now, substitute $x_1$, $y_1$, and $m$ into the point-slope formula: $$ y - 4 = \frac{1}{2}(x - 2) $$
- Simplify the equation
Distribute $\frac{1}{2}$ on the right side of the equation: $$ y - 4 = \frac{1}{2}x - 1 $$
Now, add 4 to both sides to solve for $y$: $$ y = \frac{1}{2}x + 3 $$
The equation of the line parallel to $y = \frac{1}{2}x$ that passes through the point (2, 4) is $y = \frac{1}{2}x + 3$.
More Information
The new line has the same slope as the original line, indicating they will never intersect. The y-intercept of the new line is 3, which means it will cross the y-axis at the point (0, 3).
Tips
- Forgetting that parallel lines have the same slope. Ensure you take the slope from the given line correctly.
- Misapplying the point-slope formula. It’s important to correctly plug in the point and slope into the formula.
- Failing to simplify the equation correctly can lead to incorrect results.
