Complete The Equation Of The Line Through $(2,-2)$ And $(4,1)$. Use Exact Numbers.$y = \square$

Feb 25, 2025 by ADMIN 100 views

**Complete the Equation of the Line through Two Points** =====================================================

Introduction

In mathematics, the equation of a line can be determined using the slope-intercept form, which is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. However, when we are given two points on the line, we can use the two-point form to find the equation of the line. In this article, we will discuss how to complete the equation of the line through two points.

The Two-Point Form

The two-point form of a line is given by the equation:

y - y1 = m(x - x1)

where (x1, y1) and (x2, y2) are the two given points on the line, and m is the slope of the line.

Finding the Slope

To find the slope of the line, we can use the formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are the two given points on the line.

Example

Let's say we are given two points (2, -2) and (4, 1). We can use the two-point form to find the equation of the line.

Step 1: Find the Slope

First, we need to find the slope of the line using the formula:

m = (y2 - y1) / (x2 - x1)

Plugging in the values, we get:

m = (1 - (-2)) / (4 - 2) m = (1 + 2) / 2 m = 3 / 2

Step 2: Find the Equation of the Line

Now that we have the slope, we can use the two-point form to find the equation of the line:

y - y1 = m(x - x1)

Plugging in the values, we get:

y - (-2) = (3/2)(x - 2)

Step 3: Simplify the Equation

To simplify the equation, we can multiply both sides by 2 to get rid of the fraction:

2(y - (-2)) = 2((3/2)(x - 2)) 2y + 4 = 3(x - 2)

Step 4: Expand and Simplify

Expanding and simplifying the equation, we get:

2y + 4 = 3x - 6 2y = 3x - 10

Step 5: Write the Equation in Slope-Intercept Form

Finally, we can write the equation in slope-intercept form by isolating y:

y = (3/2)x - 5

Q&A

Q: What is the two-point form of a line?

A: The two-point form of a line is given by the equation y - y1 = m(x - x1), where (x1, y1) and (x2, y2) are the two given points on the line, and m is the slope of the line.

Q: How do I find the slope of a line using the two-point form?

A: To find the slope of a line using the two-point form, you can use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the two given points on the line.

Q: Can I use the two-point form to find the equation of a line if I only have one point?

A: No, you cannot use the two-point form to find the equation of a line if you only have one point. You need at least two points to find the equation of a line.

Q: How do I simplify the equation of a line using the two-point form?

A: To simplify the equation of a line using the two-point form, you can multiply both sides of the equation by a common factor to get rid of any fractions, and then expand and simplify the equation.

Q: Can I use the two-point form to find the equation of a line if the points are not on the same line?

A: No, you cannot use the two-point form to find the equation of a line if the points are not on the same line. The two-point form only works for lines that pass through two given points.

Conclusion

In this article, we discussed how to complete the equation of the line through two points using the two-point form. We also answered some common questions about the two-point form and how to use it to find the equation of a line.