The ordinate of a point lying on the line joining the points (6, 4) and (7, -5) is -23. Find the co-ordinates of that point. 

Answer

To find the coordinates of the point lying on the line joining the points (6, 4) and (7, -5) with a given ordinate of -23, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given points**: We have two points A(6, 4) and B(7, -5).

2. **Find the slope of the line**: The slope (m) of the line joining two points (x1, y1) and (x2, y2) is given by the formula:
\[ m = \frac{y2 - y1}{x2 - x1} \]
Here, \(y1 = 4\), \(y2 = -5\), \(x1 = 6\), and \(x2 = 7\).
\[ m = \frac{-5 - 4}{7 - 6} = \frac{-9}{1} = -9 \]

3. **Use the point-slope form to find the equation of the line**: The equation of the line can be written as:
\[ y - y1 = m(x - x1) \]
Substituting the values:
\[ y - 4 = -9(x - 6) \]

4. **Simplify the equation**: Distributing the slope:
\[ y - 4 = -9x + 54 \]
Rearranging gives:
\[ y = -9x + 54 + 4 \]
\[ y = -9x + 58 \]

5. **Convert to standard form**: Rearranging gives:
\[ 9x + y = 58 \]

6. **Substitute the given ordinate**: We know the ordinate (y-coordinate) is -23. Substitute y = -23 into the equation:
\[ 9x + (-23) = 58 \]

7. **Solve for x**: Rearranging gives:
\[ 9x - 23 = 58 \]
Adding 23 to both sides:
\[ 9x = 58 + 23 \]
\[ 9x = 81 \]
Dividing by 9:
\[ x = \frac{81}{9} = 9 \]

8. **Find the coordinates**: The coordinates of the point are:
\[ (x, y) = (9, -23) \]

### Final Answer:
The coordinates of the point are (9, -23).

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