https://file.aiquickdraw.com/mathbot/file/173027677370196io6dfe.jpg

To find the equation of a line that passes through two given points, we can use the point-slope form of the line equation, \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.

The slope \( m \) is calculated as:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Let's go through each pair of points:

---

### (a) Points: \( (0, 0) \) and \( (3, -12) \)

1. **Calculate the slope \( m \):**
   \[
   m = \frac{-12 - 0}{3 - 0} = \frac{-12}{3} = -4
   \]

2. **Equation of the line:**  
   Using the point \( (0, 0) \), the equation is:
   \[
   y = -4x
   \]

---

### (b) Points: \( (-4, 2) \) and \( (0, 0) \)

1. **Calculate the slope \( m \):**
   \[
   m = \frac{0 - 2}{0 - (-4)} = \frac{-2}{4} = -\frac{1}{2}
   \]

2. **Equation of the line:**  
   Using the point \( (0, 0) \), the equation is:
   \[
   y = -\frac{1}{2}x
   \]

---

### (c) Points: \( (1, 1) \) and \( (4, 4) \)

1. **Calculate the slope \( m \):**
   \[
   m = \frac{4 - 1}{4 - 1} = \frac{3}{3} = 1
   \]

2. **Equation of the line:**  
   Using the point \( (1, 1) \), the equation is:
   \[
   y = x
   \]

---

### (d) Points: \( (-5, 3) \) and \( (1, -9) \)

1. **Calculate the slope \( m \):**
   \[
   m = \frac{-9 - 3}{1 - (-5)} = \frac{-12}{6} = -2
   \]

2. **Equation of the line:**  
   Using the point \( (-5, 3) \), the equation is:
   \[
   y = -2x - 7
   \]

---

### Summary of Equations

- **(a)** \( y = -4x \)
- **(b)** \( y = -\frac{1}{2}x \)
- **(c)** \( y = x \)
- **(d)** \( y = -2x - 7 \)

Would you like further details on any of these calculations?

---

#### Related Questions
1. How do you find the slope when given two points?
2. What is the point-slope form of a line equation?
3. How do you find the y-intercept if you know the slope and one point?
4. How would the equation change if the points were reversed?
5. How can you verify that a point lies on a specific line?

#### Tip
For any two points, you can always calculate the slope to determine if the line is increasing, decreasing, or horizontal.

Find the equation of the line joining these pairs of points: (a) (0, 0) and (3, -12), (b) (-4, 2) and (0, 0), (c) (1, 1) and (4, 4), (d) (-5, 3) and (1, -9).

Learn how to find the equation of a line passing through pairs of points with a detailed step-by-step solution. This problem involves calculating the slope, using point-slope form, and is suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation