To find the equation of a line that passes through a given point and is parallel to another line, you need a couple of important pieces of information: the slope of the original line and the coordinates of the point through which the new line must pass.
1. **Identify the slope of the given line**: The line is represented by the equation y = 2x + 1. In this equation, the slope (m) is the coefficient of x, which is 2.
2. **Understand parallel lines**: Parallel lines have the same slope. This means that the slope of the new line we are looking for is also 2.
3. **Use the point-slope form**: The point-slope form of a line’s equation is given as: y – y1 = m(x – x1), where (x1, y1) is a point on the line and m is the slope. Here, our point is (1, 4), so x1 = 1 and y1 = 4.
4. **Substitute the values into the equation**: Replacing m, x1, and y1 in the point-slope equation gives us:
y – 4 = 2(x – 1)
5. **Simplify the equation**: Now, we simplify this equation to find the slope-intercept form (y = mx + b):
- y – 4 = 2x – 2
- y = 2x + 2
6. **Final equation**: Therefore, the equation of the line passing through the point (1, 4) and parallel to the line y = 2x + 1 is y = 2x + 2.
This newly formed line has the same slope as the original line and will run parallel through the specified point!