For what values of a and b points `(1,1), ( 2, 3) (3,a) and (b,7)` are collinear.

Answer

To determine the values of \( a \) and \( b \) for which the points \( (1, 1) \), \( (2, 3) \), \( (3, a) \), and \( (b, 7) \) are collinear, we will use the concept of slopes. Points are collinear if the slopes between any two pairs of points are equal.

### Step-by-Step Solution:

1. **Identify the Points**:
Let the points be:
- \( A(1, 1) \)
- \( B(2, 3) \)
- \( C(3, a) \)
- \( D(b, 7) \)

2. **Calculate the Slope of Line Segment AB**:
The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For points \( A(1, 1) \) and \( B(2, 3) \):
\[ m_{AB} = \frac{3 - 1}{2 - 1} = \frac{2}{1} = 2 \]

3. **Set Up the Slope of Line Segment BC**:
Now, we need to find the slope of line segment \( BC \) which connects points \( B(2, 3) \) and \( C(3, a) \):
\[ m_{BC} = \frac{a - 3}{3 - 2} = a - 3 \]
Since the points are collinear, we set the slopes equal:
\[ m_{AB} = m_{BC} \]
Thus,
\[ 2 = a - 3 \]

4. **Solve for \( a \)**:
Rearranging the equation gives:
\[ a = 2 + 3 = 5 \]

5. **Calculate the Slope of Line Segment CD**:
Next, we calculate the slope of line segment \( CD \) which connects points \( C(3, a) \) and \( D(b, 7) \):
\[ m_{CD} = \frac{7 - a}{b - 3} \]
Again, since the points are collinear, we set the slopes equal:
\[ m_{BC} = m_{CD} \]
Thus,
\[ a - 3 = \frac{7 - a}{b - 3} \]

6. **Substitute the Value of \( a \)**:
We already found \( a = 5 \). Substituting this value into the equation gives:
\[ 5 - 3 = \frac{7 - 5}{b - 3} \]
Simplifying this:
\[ 2 = \frac{2}{b - 3} \]

7. **Solve for \( b \)**:
Cross-multiplying gives:
\[ 2(b - 3) = 2 \]
Dividing both sides by 2:
\[ b - 3 = 1 \]
Thus,
\[ b = 1 + 3 = 4 \]

### Final Values:
The values of \( a \) and \( b \) are:
- \( a = 5 \)
- \( b = 4 \)