How to Find the Equation of the Line Passing Through Given Points

When you need to find the equation of a line that passes through two given points, there are several methods to achieve this. Each method ultimately relies on the equation of a line in slope-intercept form, y mx b, where m is the slope, and b is the y-intercept. Let's explore these methods using an example of finding the equation of the line passing through points (2, 3) and (1, 5).

Slope-Intercept Method

The first method involves using the slope-intercept form of a line equation. This method is straightforward and relies on the formula:

Find the slope m between the points. The formula for slope is: m y2 - y1 / x2 - x1. Substitute the slope and one of the points into the slope-intercept form y mx b and solve for b.

In the given example with points (2, 3) and (1, 5), the slope m is calculated as:

m (5 - 3) / (1 - 2) -2 / -1 2

Now, using the point (2, 3) in the equation y mx b:

3 2(2) b

Solving for b:

3 4 b

b -1

Thus, the equation of the line is:

Point-Slope Method

The second method is the point-slope method. This method starts by defining the coordinates properly.

Identify the coordinates. In this case, A(2, 3) and B(1, 5). Use the formula: y - y1 m(x - x1) to find the equation of the line.

Using the slope m -2 and the point (2, 3) in the equation:

y - 3 -2(x - 2)

Expanding this:

y - 3 -2x 4

Adding 3 to both sides:

y -2x 7

Algebraic Method

The third method involves a bit more algebraic manipulation. Start with the equation y mx b.

Substitute the points (2, 3) and (1, 5) into the equation: 3 2m b 5 m b Now, solve the system of equations. Subtract the second equation from the first: 3 - 5 2m - m b - b -2 m Substitute m -2 back into one of the original equations to find b: 3 2(-2) b 3 -4 b b 7

The final equation of the line is:

y -2x 7

Conclusion

The method you choose to find the equation of the line passing through two points will depend on your comfort level with each approach. Each method utilizes the fundamental principle of the slope-intercept form, y mx b, and helps you determine the equation of the line that passes through the given points. No matter which method you use, understanding how each step works will make your homework or any similar problem much more approachable.

Frequently Asked Questions (FAQ)

1. What is the slope-intercept form of a line equation?

The slope-intercept form is a way of representing a line on a graph, defined as y mx b, where m is the slope and b is the y-intercept. This form is very useful for finding the equation of a line given its slope and y-intercept.

2. How do you find the slope of a line?

To find the slope of a line passing through two points (x1, y1) and (x2, y2), use the formula: m (y2 - y1) / (x2 - x1).

3. Can you explain the point-slope form of a line equation?

The point-slope form of a line equation is y - y1 m(x - x1), where y1 and x1 are the coordinates of a known point on the line, and m is the slope. This form is particularly useful when you know the slope and a point on the line.