1. ## Show that gradient is m for all positions...

There are two questions that I have no idea how to solve:

1. Let P, with coordinates(p,q) be a fixed point on the curve with equation y = mx + c and let Q, with coordinates (r,s), be any other point on y = mx + c. Use the fact that the coordinates of P and Q satisfy the equation y = mx + c to show that the gradient of the PQ is m for all positions of Q.

2. There are some values of a, b, and c for which the equation ax + by + c = 0 does not represent a straight line. Give an example of such values.

2. Originally Posted by struck
There are two questions that I have no idea how to solve:

1. Let P, with coordinates(p,q) be a fixed point on the curve with equation y = mx + c and let Q, with coordinates (r,s), be any other point on y = mx + c. Use the fact that the coordinates of P and Q satisfy the equation y = mx + c to show that the gradient of the PQ is m for all positions of Q.

2. There are some values of a, b, and c for which the equation ax + by + c = 0 does not represent a straight line. Give an example of such values.
I cannot quite understand the first question. Do you mean point Q is a random point? Q is at (r,s) now, but Q may be at (t,u), or (v,w), or (x,y), or.....?

For the fixed point P,
q = m(p) +c --------------(1)

For the random point Q, which is at (r,s) now,
s = m(r) +c -------------(2)

Eq.(1) minus Eq.(2),
q -s = m(p) -m(r)
q -s = m(p -r)
m = (q-s) /(p -r) ------(i)**

If Q is at another position, say at (x,y),
y = m(x) +c ------------(3)

Eq.(1) minus Eq.(3),
q -y = m(p) -m(x)
q -y = m(p -x)
m = (q -y) /(p -x) ------(ii)**

Now, compare Eq.(i) and Eq.(ii).
The gradient m remains the same even if point Q transferred from (r,s) to (x,y).

Is that what your first question wants to see?

-------------------------------------------------------

ax +by +c = 0 --------(4)

Eq.(4) will always be a straight line as long as there is a variable ...x or y... left on the equation and a non-xero c is also left in the equation. Meaning,
ax +c = 0 is a line. ----**
by +c = 0 is a line. ----**
c = 0 is not a line.

ax = 0 is not a line
by = 0 is not a line.
However, ax +by = 0 is a line. ----**

Therefore, if any two of the a,b,c are zeros at the same time, there will be no straight line.
So, a=0, b=0, c = any number -----no straight line.
a = 0, b = any number, c = 0 -------no line.
a = any number, b=0, c=0 ----------no line.