Exercises on finding the intersection of two lines

(1) Find the intersection of L: 2x+3y=30 and K: 3x-2y=19?   Solution

 
(2) Find the intersection of L: 2x + 3y = 14  and K: 3x - 2y = -5? Solution

 
(3)  Find the intersection of L: 2x = y -5 and K: x + y = 8? Solution


(4)  Find the intersection of L: 2x - y = 7  and K: x + 2y = 11? 

 Solution


(5)  Find the intersection of L: 3x - 2y = -10 and K: 2x + 3y = 2 ? 

 Solution

 

 

 

 

 

 

 

 

 

 

 

(1) Find the intersection of L:2x+3y=30 and K:3x-2y=19?

There are two unknowns in these equations.  We must eliminate one of the unknowns. We want to get rid of the x.  The lowest common multiple of 2x and 3x is 6x.  

  *  2x + 3y = 30
     3x - 2y = 19		 multiply by 3
multiply by 2
     6x + 9y = 90
     6x - 4y  = 38     		 We must subtract these equations so we can get rid of the x's.  To do this we change the sign of the lower one and add
     6x + 9y =     90
    -6x + 4y  = - 38 
             13y = 52

  ==> y=4   We plug this 4 in for Y on top equation *
     2x + 3(4) = 30
     2x = 30 - 12 
     2x = 18  


 ==> x = 9     The point of intersection is (9,4)         Return to exam

 

(2) Find the intersection of L: 2x + 3y = 14  and K: 3x - 2y = -5?  

There are two unknowns in these equations.  We must eliminate one of the unknowns. We want to get rid of the x.  The lowest common multiple of 2x and 3x is 6x.  

  * 2x + 3y = 14
     3x - 2y =  -5		 multiply by 3
multiply by 2
     6x + 9y =   42
     6x - 4y  = -10     		 We must subtract these equations so we can get rid of the x's.  To do this we change the sign of the lower one and add
     6x + 9y =     42
    -6x + 4y  = +10 
             13y =  52

  ==> y=4   We plug this 4 in for Y on top equation *
     2x + 3(4) = 14
     2x = 14 - 12 
     2x = 2  


 ==> x = 1     The point of intersection is (1,4)            Return to exam

 

 

(3)  Find the intersection of L: 2x = y -5 and K: x + y = 8? 

There are two unknowns in these equations.  We must eliminate one of the unknowns. 

     2x = y - 5
       x + y = 8		 Change both equations into same form
   * 2x - y = -5
       x + y = 8		 We want to get rid of the X.  We can just add the equations as the X's will be cancelled
     3x  = 3
      x  = 1     		 We must subtract these equations so we can get rid of the x's.  To do this we change the sign of the lower one and add.  We plug  1  in for x on top equation *
     2(1) - y = -5
     2 - y = -5 
      -y = -5 -2
       -y = -7
         y = 7  


 ==> y = 7     The point of intersection is (1,7)         Return to exam

 

(4)  Find the intersection of L: 2x - y = 7  and K: x + 2y = 11? 

There are two unknowns in these equations.  We must eliminate one of the unknowns. We want to get rid of the y.  The lowest common multiple of 1y and 2y is 2y.  

  *  2x -  y =   7
       x + 2y = 11		 multiply by 2
     4x - 2y = 14
      x  +2y = 11     		 We must add these equations so we can get rid of the y's.  To do this we change the sign of the lower one and add
     5x  = 25 ==> x=5   We plug this 5 in for X on top equation *
     2(5) - y = 7
     -y = 7 - 10 
     -y = -3  


 ==> y = 3     The point of intersection is (5,3)         Return to exam

 

(5)  Find the intersection of L: 3x - 2y = -10 and K: 2x + 3y = 2 ? 

There are two unknowns in these equations.  We must eliminate one of the unknowns. We want to get rid of the y.  The lowest common multiple of 2y and 3y is 6y. 

  *  3x - 2y = -10
      2x + 3y =  2		 multiply by 3
multiply by 2
     9x -  6y = -30
     4x + 6y  =  4     		We must subtract these equations so we can get rid of the x's.  To do this we change the sign of the lower one and add
     13x = -26
    
  ==> x=-2  We plug  -2 in for X on top equation *
     3(-2) -2y  = -20
     -2y = -20 + 6
     -2y = -14 


 ==> y = 7     The point of intersection is (-2,7)         Return to exam