How to Write an Equation for a Line Parallel to a Given Line That Contains a Point?

To find the equation of a line parallel to a given line and containing a specific point, follow these steps:

1. Identify the slope of the given line: If you have the equation of the given line, express it in the slope-intercept form, which is y = mx + b, where m is the slope.
2. Use the same slope for the new line: Lines that are parallel share the same slope. Thus, the slope of your new line will be equal to the slope of the given line.
3. Substitute the slope and the point into the point-slope form: Use the point-slope form of the equation for a line, which is y – y_1 = m(x – x_1), where (x_1, y_1) is the point through which the new line passes, and m is the slope you identified earlier.
4. Simplify to find the equation: Rearranging the point-slope form will give you the equation of the new line.

Example: Suppose the given line is y = 2x + 3, and you want to find the equation of the line parallel to it that passes through the point (1, 4). The slope of the given line is 2. Thus, using the point-slope form:

y - 4 = 2(x - 1)

Simplifying this leads to:

y - 4 = 2x - 2 
=> y = 2x + 2

So, the equation of the line parallel to the given line and containing point (1, 4) is y = 2x + 2.