Write Equation For Line L In Point Slope Form

Quick Links :

Finding the equation of a line is a fundamental concept in mathematics, particularly in geometry and algebra. When we're given a point and the slope of a line, one of the most straightforward ways to express the line's equation is by using the point-slope form. This form is especially useful because it allows us to directly use the information we have about the line's slope and a point it passes through.

What is the Point-Slope Form?

The point-slope form of a line's equation is given by:

[y - y_1 = m(x - x_1)]

Where:

• (m) is the slope of the line.
• ((x_1, y_1)) is a point on the line.

This formula is derived from the slope formula, (m = \frac{y_2 - y_1}{x_2 - x_1}), by rearranging it to solve for (y), and then using the point ((x_1, y_1)) as a reference.

How to Write the Equation of Line L in Point-Slope Form?

Let's say we have a line (L) and we know its slope (m) and a point ((x_1, y_1)) through which it passes. To write the equation of (L) in point-slope form, follow these steps:

1. Identify the Slope and the Point: Make sure you have the slope (m) and a point ((x_1, y_1)) on the line.

2. Plug the Values into the Formula: Substitute the values of (m), (x_1), and (y_1) into the point-slope formula: (y - y_1 = m(x - x_1)).

3. Simplify if Necessary: Depending on the values you've plugged in, you might need to simplify the equation. This could involve simplifying fractions or rearranging terms, but it's often useful to leave the equation in the point-slope form for clarity.

Example

Let's say we're given that the slope of line (L) is (m = 2) and it passes through the point ((3, 4)).

1. We identify (m = 2), (x_1 = 3), and (y_1 = 4).

2. We substitute these values into the point-slope formula: [y - 4 = 2(x - 3)]

3. Simplifying isn't strictly necessary here, as the equation is already clear. However, if we wanted to express it in slope-intercept form ((y = mx + b)), we could expand and rearrange: [y - 4 = 2x - 6] [y = 2x - 2]

But for many purposes, the point-slope form is perfectly suitable and clearly communicates the slope and a point on the line.

Conclusion

Writing the equation of a line in point-slope form is a straightforward process once you have the slope and a point on the line. This form is particularly useful in algebra and geometry for solving problems involving lines. By following the simple steps outlined above, you can easily express the equation of any line for which you have the necessary information.

Frequently Asked Questions