solve by substitution give a geometric interpertation of the system and its solution

Question

3x-2y=22

5x+5y=-15

Answer 1

If you want to give a geometric interpretation you might want to graph the equations (or at least know what the graphs look like).  So put each equation in slope-intercept form y = mx + b  

 

3x - 2y = 22  ==>  2y = 3x - 22  ==>  y = (3/2)x - 11

5x + 5y = -15  ==>  5y = -5x - 15 ==>  y = -x - 3 

 

Solve the system by setting the two values for y equal to each other  

 

(3/2)x - 11 = -x - 3

(5/2)x = 8

x = 8*(2/5) = 16/5 

 

Plug this value for x in to the 5x + 5y = -15 to solve for y 

 

5*(16/5) + 5*y = -15

16 + 5*y = -15

5*y = -31

y = -31/5 

 

The geometric interpretation is that there are 2 straight lines with unequal slopes so they intersect at a single point which is (16/5,-31/5)  

 

Check:  3(16/5) - 2(-31/5) = 48/5 + 62/5 = 110/5 = 22

            5(16/5) + 5(-31/5) = 16 - 31 = -15