Hi, can anyone help me out with solving this equation.

Question

Express the lines given by px + 5y = q and 3x − qy = 5q, (q ≠ 0), in
the y = mx + c form.
b Hence, determine the values of p and q so the system of equations
px + 5y = q
3x − qy = 5q
will have infinitely many solutions.
c What relationship must exist between p and q so the lines px + 5y = q and
3x − qy = 5q will intersect?

Answer 1

 

Part A:

 

px + 5y = q

 

5y = q - px

 

y = (-p/5)x + (q/5)

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3x - qy = 5q

 

3x - 5q = qy

 

(3/q)x - (5q/q) = y

 

(3/q)x - 5 = y

 

Part B)

 

So in order to get infinitely many solutions, the equations must be the same:

 (3/q)x - 5 = (-p/5)x + (q/5)

 

Then 3/q = -p/5  and q/5 = -5

 

From the second equation , q = -5 * 5 = -25

Plugging this into the first equation:

    -3/25 = -p/5

 

   Multiplies both sides by -5:

     3/5 = p

 

So p=3/5 and q = -25

 

The equations then become:

 y  = (-3/5)/5x + -25/5 =

 

   (-3/25)x - 5 = (3/q)x - 5

 

 

Part C:

 THey will intersect if the slopes are different.

If the slopes are the same, then

  -p/5 = 3/q

 

 -pq = 15

 

 pq = -15

 

 So the lines will intersect unless p*q = -15