Understanding the Equation of a Line Parallel to a Given Line Through a Specific Point

When dealing with the geometric properties of lines, one frequent task is to find the equation of a line that is parallel to a given line and passes through a specific point. This article will guide you through the process step-by-step, using the example where the given line is 5x - 2y 10 and we need to find the equation of the line parallel to it passing through the point (4, 4).

The Given Line and Its Slope

The equation of the given line is 5x - 2y 10. To understand the slope of the line, we first need to convert this equation into the slope-intercept form, which is y mx b, where m is the slope and b is the y-intercept.

Convert the Given Line Equation to Slope-Intercept Form

Starting from 5x - 2y 10, we rearrange it to solve for y: -2y -5x 10 y 5/2x - 5

Thus, the slope m1 of the given line is -5/2.

Slope of the Parallel Line

Parallel lines have the same slope. Therefore, the slope m2 of the line we are trying to find is also -5/2.

Using the Point-Slope Form

Given a point (4, 4) and a slope of -5/2, we can use the point-slope form of the equation of a line, which is y - y1 m (x - x1).

Applying the Point-Slope Form

Substituting the point (4, 4) and the slope -5/2 into the point-slope form:

y - 4 -5/2 (x - 4)

Multiplying both sides of the equation by 2 to clear the fraction:

2 (y - 4) -5 (x - 4) 2y - 8 -5x 20

Adding 5x and 8 to both sides to rearrange the equation into standard form:

5x 2y 28

Verification of the Parallel Line

As verified, the equation 5x - 2y 10 and 5x 2y 28 are indeed parallel lines, sharing the same slope of -5/2 but with different y-intercepts.

Conclusion

By following these steps, we can easily find the equation of a line that is parallel to a given line and passes through a specified point. The detailed process and the derived equation for this specific problem are as follows:

The line parallel to 5x - 2y 10 and passing through the point (4, 4) is 5x 2y 28.

This method can be applied to any such problem by ensuring the slope is the same and then substituting the given point into the point-slope form.