Given the points A (1,8), B(4,7) and O(0,0), find the coordinates of the point C at which these two lines intersect?

To find the coordinates of point C, we first need to find the equations of the two lines passing through points A and B.

The equation of the line passing through points A(1,8) and O(0,0) can be found using the slope-intercept form:

m = (y2 - y1) / (x2 - x1)
m = (8 - 0) / (1 - 0)
m = 8

Using point-slope form, the equation of the line passing through A and O is:
y - 8 = 8(x - 1)
y = 8x

Similarly, the equation of the line passing through points B(4,7) and O(0,0) can be found using the slope-intercept form:

m = (y2 - y1) / (x2 - x1)
m = (7 - 0) / (4 - 0)
m = 7/4

Using point-slope form, the equation of the line passing through B and O is:
y - 7 = 7/4(x - 4)
y = 7/4x + 5.25

Now, to find the coordinates of point C where these two lines intersect, we need to solve the system of equations:

y = 8x
y = 7/4x + 5.25

Setting the two equations equal to each other:
8x = 7/4x + 5.25
Multiplying by 4 to get rid of the fraction:
32x = 7x + 21
25x = 21
x = 21/25

Substitute x back into one of the equations to find y:
y = 8(21/25)
y = 168/25

Therefore, the coordinates of point C are (21/25, 168/25).

Step 1: Calculate the slope of line AB.

```
slope of AB = (7 - 8) / (4 - 1) = -1/3
```

Step 2: Use the point-slope form to write the equation of line AB.

```
y - 7 = -1/3 (x - 4)
=> y = -1/3 x + 11/3
```

Step 3: Calculate the slope of line OC.

Since line OC passes through the origin O(0,0), its slope is undefined.

Step 4: Since the two lines intersect, their slopes must be equal.

Since the slope of line OC is undefined and the slope of line AB is -1/3, line OC must be vertical.

Step 5: Find the x-coordinate of point C.

Since line OC is vertical, it passes through the x-axis at the same point as line AB. Therefore, the x-coordinate of point C is the same as the x-coordinate of point B, which is 4.

Step 6: Substituting the x-coordinate of C into the equation of line AB, find the y-coordinate of C.

```
y = -1/3 (4) + 11/3
y = 2
```

Therefore, the coordinates of the point C are (4, 2).