I really need urgent help for this question... What do they mean by determine the conditions? What conditions? And how do I do the question?

Question

Consider the simultaneous equations

px-y=6,
8x-2y=q

determine the conditions that p and q must satisfy of the simultaneous equations have
(i) an infinite number of solutions
(ii) no solution,
(iii) a unique solution, i.e. Only one solution.

 

 

 

What do they mean by determine the conditions? What do they mean by conditions? And how do I do the question?

Answer 1

First, TXH for being a thinking person and for asking this question multiple times.  Obviously, you want an answer you understand (not just an answer to turn in).

 

The words "under what conditions?" are the "if" part of an "if-then" statement.  An "if-then" phrase means that the result happens IF the conditions are met.  Sometimes, especially in math, it might be even more exclusive and be "if-and-only-if" conditions.  For example, "If you get an A on the next math test, then you get to go to on the trip to the Spy Museum (or something that you like so that it motivates you)."  Well, you can make up better examples than I can.  The "if" part are the conditions.

 

This problem gives three situations:

 

(i) an infinite number of solutions  - this happens if and only if (sometimes written 'iff') the two equations actually are the same line

(ii) no solution  - this happens iff the two lines are parallel (if they never intersect, there is no solution)

(iii) a unique solution, i.e. Only one solution  - this happens when two lines cross  (the solution is unique because the lines cross only once if they are straight lines)

 

So, what values of p and q make situation (i) true?

       px  - y=6
       8x -2y=q

  These equation are in Standard Form (leave them that way).  In order to get a common coefficient for y, multiply the first equation by 2:

       2px  - 2y = 12

        8x  -  2y = q

  For these equations to be exactly the same, 2p=8, or p=8, and q=12.

  For this system of equations to have infinite solutions (that is, be the same line), the conditions that both p=4 and q=12 must be met.

 

What situation (conditions) of p and q make (ii) true?

    Now, let's convert the equations to slope-intercept form:

         y = mx + b      {where m = slope and b = y-intercept)

         y = px - 6

         y = 4x -q/2

    The lines are parallel iff they have the same slope (that is, p=4) but they have a different y-intercept (that is, q≠12).

   For these equations to have no solutions (that is, be parallel), the conditions that both p=4 and q≠12 must be met.

 

And, what conditions of p and q make (iii) true?

   Now, some logic is important.  We can not allow p=4, ever!   Because if p=4 and q=12 it is case (i).  But, if p=4 and q≠12 it is case (ii)  --  that accounts for everything when p=4.  Thus, p can never be equal to 4.

   What can q be?    q can be any value (-∞ to +∞ as long as p≠4).

  For these equations to have exactly one solution (that is, intersect at a point), p≠4 and q can have any value.

Answer 2

 

Hi Amy,

 

It's to figure out under what conditions the set of equations satisfies the given statement above.

 

In this specific problem, we need to figure out

 

how we need to set up p and q

 

so that the set of two linear equations can meet the given statement (i), (ii), or (iii).

 

First, let's rewrite each equation as follows.

 

     px - y = 6   ⇒   y = px - 6      (of which y-intercept is -6)

 

     8x - 2y = q ⇒   y = 4x - q/2   (of which slope is 4)

 

In order to satisfy the statement (i),

 

the two equations need to be identical with each other.

 

That is, p = 4 and q = 12.

 

In order to satisfy the statement (ii),

 

the two equations need to have an equal slope, but different y-intercepts.

 

That is, p = 4 and q ≠ 12.

 

In order to satisfy the statement (iii),

 

the two equations need to have different slopes.

 

That is, p ≠ 4.

 

Hope this helps.