SOLUTION: Find the value of k if 5x + 7y = k and 10x +19y = 3 have infinite number of solutions

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Question 972322: Find the value of k if 5x + 7y = k and 10x +19y = 3 have infinite number of solutions
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
Find the value of k if 5x + 7y = k and 10x +19y = 3 have infinite
number of solutions
This is impossible. Both the slopes and the y-intercepts must be the same in
both equations in order for two equations to have an infinite number of
solutions.   

No matter what value k takes on, the slope of 5x + 7y = k is the same. To show
that they do not have the same slope, we put them both in slope-intercept
form:  "y = mx + b":

5x + 7y = k
     7y = -5x + k
    7y%2F7%22%22=%22%22expr%28%28-5%29%2F7%29x%2Bk%2F7
    y%22%22=%22%22expr%28-5%2F7%29x%2Bk%2F7

Comparing with y = mx + b the slope is -5%2F7 and the y-intercept is
the point %28matrix%281%2C3%2C0%2C%22%2C%22%2Ck%2F7%29%29

For the other equation:

10x + 19y = 3
      19y = -10x + 3
    19y%2F19%22%22=%22%22expr%28%28-10%29%2F19%29x%2B3%2F19
    y%22%22=%22%22expr%28-10%2F19%29x%2B3%2F19
 
Comparing with y = mx + b the slope is -10%2F19 and the y-intercept is
the point %28matrix%281%2C3%2C0%2C%22%2C%22%2C3%2F19%29%29

They do not have the same slope, so no matter what value of k we use,
there will always be exactly one solution.

Edwin