Intersection of two lines

Question

a(1−ax)=y
2(ax−y)+1=−3(x+y)

I need to find the Intersection of these two lines

The answer is: (1/a-3 , 3a/3-a) but I can't get to it

Answer 1

To find the intersection of these two lines, we need to solve the system of equations.  We need to simplify each equation so that they are easier to work with.

 

 

a(1 - ax) = y --------> a - a²x = y  

 

2(ax - y) + 1 = -3(x + y)  ------>  2ax - 2y + 1 = -3x - 3y  -------> y = -2ax - 3x - 1

 

 

These are the equations we will work with.

 

a - a²x = y                 eq1

 

y = -2ax - 3x -1            eq2

 

 

Notice that y is defined in both equations.  We can just equate the y's.

 

a - a²x = -2ax - 3x - 1

 

 

Move all the x terms to one side of the equation.

 

a + 1 = a²x - 2ax - 3x

 

a + 1 = x(a² - 2a - 3)

 

(a + 1) / (a - 3)(a + 1) = x

 

1 / (a - 3) = x

 

 

Substitute this value of x into eq1 to solve for y.  This is because eq1 is simpler.

 

 

y = a - (a² / (a - 3))

 

y = (a(a - 3) - a²) / (a - 3)

 

y = (a² - 3a - a²) / (a - 3)

 

y = 3a / (a - 3)