Question 972322
Find  the  value  of  k  if  5x + 7y = k  and  10x +19y = 3 have  infinite
number  of  solutions
<pre>
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/7}}}{{{""=""}}}{{{expr((-5)/7)x+k/7}}}
    {{{y}}}{{{""=""}}}{{{expr(-5/7)x+k/7}}}

Comparing with y = mx + b the slope is {{{-5/7}}} and the y-intercept is
the point {{{(matrix(1,3,0,",",k/7))}}}

For the other equation:

10x + 19y = 3
      19y = -10x + 3
    {{{19y/19}}}{{{""=""}}}{{{expr((-10)/19)x+3/19}}}
    {{{y}}}{{{""=""}}}{{{expr(-10/19)x+3/19}}}
 
Comparing with y = mx + b the slope is {{{-10/19}}} and the y-intercept is
the point {{{(matrix(1,3,0,",",3/19))}}}

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

Edwin</pre>