solve for the linear function (y=mx+b) passing through points: (-2,5) and (1,-4)

To find the equation of the linear function \( y = mx + b \) that passes through the points \( (-2, 5) \) and \( (1, -4) \), we need to follow these steps:

### Step 1: Find the slope \( m \)
The slope \( m \) is given by the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Here, \( (x_1, y_1) = (-2, 5) \) and \( (x_2, y_2) = (1, -4) \).

Substituting the values:
\[
m = \frac{-4 - 5}{1 - (-2)} = \frac{-4 - 5}{1 + 2} = \frac{-9}{3} = -3
\]

### Step 2: Find the y-intercept \( b \)
Now that we have the slope \( m = -3 \), we can use one of the points to solve for \( b \). Let’s use the point \( (-2, 5) \) and substitute it into the equation \( y = mx + b \).

Substituting \( y = 5 \), \( m = -3 \), and \( x = -2 \):
\[
5 = -3(-2) + b
\]
\[
5 = 6 + b
\]
\[
b = 5 - 6 = -1
\]

### Step 3: Write the equation
Now that we have the slope \( m = -3 \) and the y-intercept \( b = -1 \), the equation of the line is:
\[
y = -3x - 1
\]

### Final Answer:
The equation of the line passing through the points \( (-2, 5) \) and \( (1, -4) \) is:
\[
y = -3x - 1
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you find the slope of a line given two points?
2. What does the slope represent in a linear function?
3. How would the equation change if the points were reversed?
4. Can you find the x-intercept for the equation \( y = -3x - 1 \)?
5. What is the general form of a linear equation, and how does it relate to the slope-intercept form?

**Tip:** When finding the slope, make sure to subtract the coordinates in the same order to avoid mistakes!

Learn how to find the equation of a linear function passing through the points (-2, 5) and (1, -4) using the slope-intercept form. This step-by-step solution covers the slope formula and solving for the y-intercept. Suitable for students in Grades 8-10.

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