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plz help with systems

 

For certain values of k and m, the system

3a + 2b = 2

6a + 2b = k + 3a + mb

has infinitely many solutions (a,b).  What are k and m?

 Jul 12, 2023
 #1
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To have something that has infinite solutions we need both sides of a equation to be the exact same. Now lets use this to our advantage. 

 

Let us simply the system of equations via substitution : 

 

3a + 2b = 2 

 

 = 3a = 2 - 2b

Now we can substitute that into the second equation : 

 

6a + 2b = k + 3a + mb 

 

= (2 - 2b) x 2 + 2b = k + ( 2 - 2b ) + mb 

 

= 4 - 4b + 2b = k + 2 -2b + mb 

 

= 4 - 2b  =  k + 2-2b + mb 

 

Now after this, we will start moving variables : 

 

4 = 2 + mb + k

 

2 = mb + k 

 

Now you may feel stuck at this point but we will need to use guess and check to find the answer : 

 

Since 2 is such a small number I would assume in this problem that some variable would be equal to 0

 

Using substituion we can find the m = 0 and k = 2 

 

Hope this helps!smiley

 Jul 12, 2023

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