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write the equation in slope intercept form for the following

Let's solve both problems by writing the equations in slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

### 8) Based on the Table of Values:
We need to calculate the slope \( m \) using two points from the table. The formula for slope is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Let's choose the points \((6, 14)\) and \((9, 12)\):
\[
m = \frac{12 - 14}{9 - 6} = \frac{-2}{3} = -\frac{2}{3}
\]

Now, substitute the slope \( m = -\frac{2}{3} \) into the slope-intercept form \( y = mx + b \). We can use one point from the table, say \((6, 14)\), to find \( b \).

Substitute \( x = 6 \), \( y = 14 \), and \( m = -\frac{2}{3} \) into the equation \( y = mx + b \):

\[
14 = -\frac{2}{3}(6) + b
\]
\[
14 = -4 + b
\]
\[
b = 18
\]

Thus, the equation is:
\[
y = -\frac{2}{3}x + 18
\]

### 9) Based on the Graph:
From the graph, we can visually determine the slope and y-intercept. The line passes through the points \((-6, 0)\) and \((0, 4)\).

The slope \( m \) is:
\[
m = \frac{4 - 0}{0 - (-6)} = \frac{4}{6} = \frac{2}{3}
\]

The y-intercept \( b \) is where the line crosses the y-axis, which is \( b = 4 \).

Thus, the equation is:
\[
y = \frac{2}{3}x + 4
\]

### Final Equations:
1. For the table (problem 8): \( y = -\frac{2}{3}x + 18 \)
2. For the graph (problem 9): \( y = \frac{2}{3}x + 4 \)

Would you like further explanation on these solutions or any other questions?

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Here are five related questions for deeper understanding:
1. How do you find the slope when given two points?
2. Why does the y-intercept represent the value of \( y \) when \( x = 0 \)?
3. Can a line have a slope of zero? If so, what does that line look like?
4. What is the significance of a negative slope compared to a positive slope?
5. How would the equation change if the line passed through the origin?

**Tip:** Always remember that the slope measures the steepness of a line. A positive slope means the line rises as it moves right, while a negative slope means it falls.

Write the equation in slope-intercept form for the following

Learn how to write linear equations in slope-intercept form (y = mx + b) based on a table of values and a graph. Includes step-by-step solutions for calculating the slope and y-intercept. Suitable for students in Grades 8-10.

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