To solve this problem, we need to determine the slope of the line passing through the given points. Here's the step-by-step approach:

Slope Formula:

The slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Step 1: Analyze the Given Points

The graph shows several points:

• Point 1: $(1, 3)$
• Point 2: $(a, 6)$
• Point 3: $(5, b)$
• Point 4: $(9, 7)$

All points lie on the same straight line, which means the slope calculated using any two pairs of points should be equal.

Step 2: Use Two Known Points to Find the Slope

Using points $(1, 3)$ and $(9, 7)$: $m = \frac{7 - 3}{9 - 1} = \frac{4}{8} = \frac{1}{2}$

Thus, the slope of the line is $\frac{1}{2}$.

Step 3: Find Missing Coordinates

Now that the slope is $\frac{1}{2}$, we use it to find the missing coordinates $a$ and $b$.

1. Find $a$: Using points $(1, 3)$ and $(a, 6)$, the slope is: $\frac{6 - 3}{a - 1} = \frac{1}{2}$ Simplify: $\frac{3}{a - 1} = \frac{1}{2}$ Cross-multiply: $2 \cdot 3 = 1 \cdot (a - 1)$ $6 = a - 1$ $a = 7$

2. Find $b$: Using points $(1, 3)$ and $(5, b)$, the slope is: $\frac{b - 3}{5 - 1} = \frac{1}{2}$ Simplify: $\frac{b - 3}{4} = \frac{1}{2}$ Cross-multiply: $2 \cdot (b - 3) = 1 \cdot 4$ $2b - 6 = 4$ $2b = 10$ $b = 5$

Final Results:

• The slope of the line is $\frac{1}{2}$.
• Missing coordinates: $a = 7$ and $b = 5$.

Would you like a detailed explanation of any specific step? Let me know!

Related Questions:

1. How do you verify that all points lie on the same line?
2. What is the equation of the line in slope-intercept form?
3. Can the slope be undefined for certain lines?
4. How would the slope change if the points were arranged differently?
5. What are some real-world applications of finding slopes?

Tip: Always use the slope formula systematically to avoid calculation errors.