Show that the three points (5, 1), (1, -1) and (11, 4) lie on a straight line. Further find
its intercepts on the axes

Answer

To show that the three points (5, 1), (1, -1), and (11, 4) lie on a straight line, we can use the two-point form of the equation of a line. We will also find the intercepts of the line on the axes.

### Step 1: Use the two-point form to find the equation of the line

The two-point form of the equation of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) \]

Let's take the points \((5, 1)\) and \((1, -1)\). Here, \(x_1 = 5\), \(y_1 = 1\), \(x_2 = 1\), and \(y_2 = -1\).

Substituting these values into the equation:

\[ y - 1 = \frac{-1 - 1}{1 - 5}(x - 5) \]

Calculating the slope:

\[ y - 1 = \frac{-2}{-4}(x - 5) \]

This simplifies to:

\[ y - 1 = \frac{1}{2}(x - 5) \]

### Step 2: Simplify the equation

Now, we can simplify the equation further:

\[ y - 1 = \frac{1}{2}x - \frac{5}{2} \]

Adding 1 to both sides:

\[ y = \frac{1}{2}x - \frac{5}{2} + 1 \]

This simplifies to:

\[ y = \frac{1}{2}x - \frac{3}{2} \]

### Step 3: Rearranging to standard form

To convert this into standard form \(Ax + By + C = 0\):

\[ \frac{1}{2}x - y - \frac{3}{2} = 0 \]

Multiplying through by 2 to eliminate the fraction:

\[ x - 2y - 3 = 0 \]

Thus, the equation of the line is:

\[ x - 2y = 3 \]

### Step 4: Verify if the third point lies on the line

Now, we need to check if the point \((11, 4)\) satisfies this equation. Substitute \(x = 11\) and \(y = 4\):

\[ 11 - 2(4) = 11 - 8 = 3 \]

Since the left-hand side equals the right-hand side, the point \((11, 4)\) lies on the line.

### Conclusion for the points

Thus, the points (5, 1), (1, -1), and (11, 4) all lie on the same straight line.

### Step 5: Find the intercepts on the axes

**X-intercept**: Set \(y = 0\) in the equation \(x - 2y = 3\):

\[ x - 2(0) = 3 \implies x = 3 \]

So, the x-intercept is \(3\).

**Y-intercept**: Set \(x = 0\) in the equation \(x - 2y = 3\):

\[ 0 - 2y = 3 \implies -2y = 3 \implies y = -\frac{3}{2} \]

So, the y-intercept is \(-\frac{3}{2}\).

### Final Result

The intercepts of the line are:
- X-intercept: \(3\)
- Y-intercept: \(-\frac{3}{2}\)