Write down an equation in the form

y = mx + b

for the line through the points (1,2) and (7,-1).

Find b

To find the equation of the line in the form \( y = mx + b \) that passes through the points \( (1, 2) \) and \( (7, -1) \), we can follow these steps:

### Step 1: Find the slope \( m \)
The slope \( m \) of the line is given by the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the given points \( (x_1, y_1) = (1, 2) \) and \( (x_2, y_2) = (7, -1) \):
\[
m = \frac{-1 - 2}{7 - 1} = \frac{-3}{6} = -\frac{1}{2}
\]

### Step 2: Use the point-slope form to find the y-intercept \( b \)
The equation of the line can now be written as:
\[
y = -\frac{1}{2}x + b
\]
To find \( b \), substitute one of the points into the equation. Let's use the point \( (1, 2) \):
\[
2 = -\frac{1}{2}(1) + b
\]
\[
2 = -\frac{1}{2} + b
\]
Add \( \frac{1}{2} \) to both sides to solve for \( b \):
\[
b = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}
\]

### Final Equation
The equation of the line is:
\[
y = -\frac{1}{2}x + \frac{5}{2}
\]

So, the value of \( b \) is \( \frac{5}{2} \).

Would you like more details or have any questions? Here are some related questions to consider:

1. How do you calculate the slope if given different points?
2. How can you convert the equation to standard form?
3. What is the significance of the y-intercept in a linear equation?
4. How would the equation change if the points were reversed?
5. How do you interpret the slope in a real-world context?

**Tip:** Remember, the slope \( m \) represents the rate of change of \( y \) with respect to \( x \), and the y-intercept \( b \) is the value of \( y \) when \( x = 0 \).

Write down an equation in the form y = mx + b for the line through the points (1,2) and (7,-1). Find b.

Discover how to find the equation of a line in slope-intercept form through the points (1,2) and (7,-1). This problem focuses on key algebraic concepts such as calculating the slope and finding the y-intercept. Perfect for students in Grades 8-10.

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