System of equation math

Question

hey I don't have idea how to calculate this problem.

please give me an advice.

kx-3y=4

4x-5y=7

x and y are variables,k is a constant .

For what value of k will system of equations have no solution ?

Answer 1

The equations are written in Standard Form (Ax+By=C).  You may, once and forever, convert that to slope intercept form (y=mx+b) and remember that the slope, m, is (-A/B).

 

For the system to have "no solution," the slope of the lines must be the same (that is, they never intersect).  So,

   k/3 = 4/5

    k = 12/5

 

So,    (12/5)x - 3y = 4     or   12x -15y = 20

and            4x - 5y = 7     or   12x -15y = 21

     are parallel.

Answer 2

k x - 3 y = 4

   k x - 4 = 3 y

          y  =k/3 x - 4/3  this  is equation 1

 

4 x -5 y = 7 

  4 x - 7 = 5 y

          y = 4/5 x - 7/5    this is equation 2

combine equation 1 and equation 2, we will get

 

          k/3 x - 4/3 = 4/5 x - 7/5

       k/3 x - 4/5 x = 4/3 - 7/5

       (k/3 -4/5) x  = 20/15 -21/15

(5k/15 - 12/15) x = -1/15

                        x = -1/(5k - 12)

 

 the denominator can't be 0, so if 5k - 12 =0, it will have no solution

 

5k -12= 0

        k= 12/5 

 

 

k is 12/5 will system of equations have no solution

Answer 3

In order a SOE to be inconsistent, the slope needs to be exactly the same so that the 2 lines are parallel and never intersect.

 

Step 1: Put equation 2 in slope-intercept form.

 

4x-5y=7

4x-7=5y

4/5 x - 7/5 = y

 

Step 2: Put eq 1 in SI form.

 

kx - 3y=4

kx-4=3y

k/3 x - 4/3 =y

 

Step 3: Set the slope of eq 1 = slope of eq 2.

 

k/3 = 4/5

5*k = 3*4

k = 12/5

 

Step 4: Check your work. Plug in k and see if the slopes are equal and intercepts are different.

 

(12/5)/3 x - 4/3 !=(does not equal) 4/5 x - 7/5

12/15 x - 20/15  != 4/5 x - 21/15

4/5 x - 20/15 != 4/5 x - 21/15

 

So the answer is k = 12/5.