Two Point Form

Two-Point Form: Definition

There are multiple ways to find the equation of a line based on the given information. The two-point form is used for deriving the equation of a line in the coordinate plane when two points of the line are given. It is a helpful way to represent a line algebraically. The line equation holds for every point on that line, meaning that each point lying on the line would satisfy the equation. Other ways to find the line equation are point-slope form, slope-intercept form, intercept form, etc. Read with us to know more about this method of finding the equation of a line.

Two-Point Form: Formula

The two-point form formula is: 

Y − y1 = (y2 − y1 / x2 − x1 )(x − x1) or, 

Y − y2 =( y2 − y1 / x2 − x1 )(x − x2)

Here, (x, y) is an arbitrary point on the line.

(x1, y1) and (x2, y2) are the coordinates of the point that lies on the line

Let us solve a couple of examples to understand the formula better. 

Example 1:  Derive the equation of the line passing through the points (-2, 3) and (3, 5). 

Here, the coordinates given to us are A (-2, 3) = (x1, y1) and B (3, 5) = (x2, y2).

Since we know the coordinates of two points lying on the line, we can easily use the two-point form calculator to find the equation of the line. 

We know that, y − y1 = (y2 − y1/ x2 −x1 )(x − x1)

By substituting the values in the given formula, we get,

y – 3 = [(5 – 3) / (3 – (-2)] (x – 2)

y – 3 = (2/5) (x – 2)

5 (y – 3) = 2 (x + 2)

5y – 15 = 2x + 4

2x – 5y + 19 = 0

It is the equation of the line passing through the coordinates (-2, 3) and (3, 5). 

Example 2: Derive the equation of the line passing through the points (1, 2) and (-1, 3). 

Here, the coordinates given to us are A(1, 2) = (x1, y1) and B(-1, 3) = (x2, y2).

Since we know the coordinates of two points lying on the line, we can easily use the two-point form calculator to find the equation of the line. 

We know that, y − y1 =( y2 − y1 / x2 −x1 )(x − x1)

By substituting the values in the given formula, we get,

y – 2 = [(3 – 2) / (-1 – 1)] (x – 1)

y – 2 = (1/(-2)) (x – 1)

Multiplying both sides of the equation (-2), we get: 

-2 (y – 2) = x – 1

-2y + 4 = x – 1

x + 2y – 5 = 0

It is the equation of the line passing through the coordinates (1, 2) and (-1, 3)

How To Derive the Two-Point Formula?

The equation of the two-point form can be easily derived if the coordinates of the two points lying on the line are given. Suppose the two fixed points lying on line are A(x1, y1) and B (x2, y2). Let us assume that there is another point C(x, y) lying on the line.  

Now, because points A, B, and C lie on the same line,  the slope of AC will be equal to the Slope of AB. Now, as per the slope formula, 

y− y1 / x − x1 = y2 − y1 / x2 − x1

Multiplying by x − x1 on both sides will give us, 

y − y1 = (y2 − y1 / x2 − x1 )(x − x1) 

This is also the equation of the two-point form formula used to derive the equation of any line of which two points are known. Similarly, the other two-point formula. 

y − y2 = (y2 − y1 / x2 − x1 )(x − x2) can be derived using the slope equation.

How To Find the Slope of a Line With the Two-Point Formula?

To find the slope of a line, we need to find the equation of a line with the help of two points lying on it. Solve the equation of the line and derive the value of y. Followed by this, compare the equation you derive with y = mx + b. In this equation, m is the line slope that can be compared and derived. 

For example: 

Derive the y-intercept of the line with the coordinates A (3, -2) and B (-1, 3) passing through it and find the slope of the line. 

Here, the coordinates given to us are A (3, -2) = (x1, y1) and B (-1, 3) = (x2, y2) 

Since we know the coordinates of two points lying on the line, we can easily use the two-point formula to find the equation of the line. 

We know that, y − y1 = (y2 − y1 / x2 −x1 )(x − x1)

Y +2 = [(3+2) / (-1 – 3)] (x – 3)

Y +2 = (5 / (-4)) / (x – 3)

Now, multiplying both sides by (-4) gives us, 

-4 (y + 2) = 5 (x – 3)

-4y – 8 = 5x – 15

-4y = 5x – 7

Y = (-5/4)x + (7/4) 

The final equation of the slope-intercept form is written as y = mx + b

Comparing our equation, we will get y-intercept, b as (7/4). 

Further, the slope of the line, m = (-5/4).

Conclusion

The two-point form is an easy way to find the equation of the line, the y-intercept, and slope. You can also use other methods to find the equation of a line if different elements such as the intercept, the slope, or both the intercepts are given. However, only the coordinates of two points of a given line are sufficient to derive the line’s equation and find the line’s slope.