State the Value of K

Date: 11/26/97 at 21:05:39
From: Jennifer
Subject: Algebra

I need an explanation and am really hopeful that I have found a good 
place for help...

State the value of k so that the given point lies on the graph of the 
given equation

A)  2x + ky = 21; (3,5)

B)  kx - 3y = 11; (-1,-5)

C)  2kx + 5y - 6 = 0; (-1,2)

Date: 12/11/97 at 11:05:49
From: Doctor Mark
Subject: Re: Algebra

Hi, Jennifer

The basic idea here is that you need to understand what it means to 
say that a pair of numbers (a,b) "solves" a linear equation, or that a 
point (a,b) lies on some graph. 

(a,b) solves a linear equation in x and y if, when you substitute a 
for x, and b for y, you end up with a statement that is true, like 
4 = 4, and not false, like 4 = 6.  

A graph of an equation in x and y is a visual way of presenting all 
the solutions to the equation. If the equation is linear, as in your 
examples, then points and solutions go hand-in-hand: 

   a point (a,b) lies on the graph if (a,b) solves the linear 
      equation, and 

   if (a,b) solves a linear equation, then the point (a,b) lies 
      on the graph.

So in your problems, saying that (for example) (3,5) lies on the 
graph of 2x + ky = 21 is the same thing as saying that the values 
x = 3, y = 5, when substituted into the given equation, give a true 
statement. Let's see where that goes:

   2(3) + k(5) = 21 --> 6 + 5k = 21

If this is to be a true statement, then k must have some value which 
makes 6 + 5k = 21 a true statement. But that is just what we mean by 
saying that k is a solution to this equation: it's a value for k which 
makes the equation true. So we just have to solve this equation for k!  
You know how to solve equations like this:

   6 + 5k = 21 --> 5k = 21 - 6 --> 5k = 15 -->  k = 15/5 = 3

So if we choose 3 for the value of k, then (3,5) solves the equation, 
and so (3,5) lies on the graph.

The other examples you gave work the same way: substitute the given 
values of x and y into the equation; this gives you an equation for k, 
which you then solve. You should, however, make sure that you use 
parentheses when you substitute. That is, it is easy to avoid "stupid" 
mistakes (which smart people often make!) if you make substitution a 
two-step process:

1. Put parentheses around the letter (variable) and *only* the   
   variable: do not include any exponents, minus signs, or 
   coefficients inside those parentheses!

2. Now write down *exactly* what you wrote in the first step, 
   including the parentheses, except instead of the variable(s) write 
    the numbers you are substituting.

Then you do the arithmetic.

For instance, in your second example, the process would look like 
this:

      kx - 3y = 11   (the equation)

  k(x) - 3(y) = 11   (put parentheses around the x and the y *only*)

k(-1) - 3(-5) = 11  (rewrite the equation on the previous line, but
                     put in the numbers)

      -k + 15 = 11  (do the arithmetic; now solve this for k)

      15 - 11 = k

            4 = k

            k = 4

Of course, you have to remember which number gets substituted for 
which letter. I find that is easy if you put the letter above the 
number, just to remind yourself, like this:

     x, y
   (-1,-5)

Then you just have to be careful.

Hope this helps, and if you have any other questions, be sure to write 
back.  

-Doctor Mark,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/