If the points (A(1, 2)), (B(4, b)), and (C(a, 6)) are collinear, find the values of (a) and (b).

To determine the values of \( a \) and \( b \) such that the points \( A(1, 2) \), \( B(4, b) \), and \( C(a, 6) \) are collinear, we can use the concept of the **slope** of a line.

The points are collinear if the slopes between any two pairs of points are equal. We will set the slopes between points \( A \) and \( B \), and between points \( B \) and \( C \), equal to each other.

### Step 1: Calculate the slope between two points.
The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

#### Slope between \( A(1, 2) \) and \( B(4, b) \):
\[
m_{AB} = \frac{b - 2}{4 - 1} = \frac{b - 2}{3}
\]

#### Slope between \( B(4, b) \) and \( C(a, 6) \):
\[
m_{BC} = \frac{6 - b}{a - 4}
\]

### Step 2: Set the slopes equal.
Since the points are collinear, the slopes must be equal:
\[
\frac{b - 2}{3} = \frac{6 - b}{a - 4}
\]

### Step 3: Solve the equation.
Cross-multiply to eliminate the denominators:
\[
(b - 2)(a - 4) = 3(6 - b)
\]

Expand both sides:
\[
b a - 4b - 2a + 8 = 18 - 3b
\]

Rearrange the terms:
\[
b a - 4b + 3b - 2a = 18 - 8
\]
\[
b a - b - 2a = 10
\]

This is the equation we have so far. Now, we need to find \( a \) and \( b \).

One way to proceed is to substitute specific values for either \( a \) or \( b \) and solve the equation, or you can attempt to isolate one variable in terms of the other.

Would you like me to continue solving for these variables, or explain further?

Solve the problem of finding the values of a and b for which points A(1, 2), B(4, b), and C(a, 6) are collinear. This solution explores algebraic methods and the concept of slope in coordinate geometry. Suitable for students in Grades 9-11.

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