Question

Find the intersection between thelines 2x + 3y = 5 and 3x + 4y = 6

Answer 1

An intersection point, is a shared coordinate. If the equations share a coordinate we can say they are "equal". There are two ways (algebraically) to find this point. Substitution or linear combination. Because the coefficients are different, let's use linear combination

First stack the equations on top of each other.

2x + 3y = 5
3x + 4y = 6

Next let's manipulate one or both of the equations to get coefficients that are equal, but opposite. For this I'm going to turn the coefficient of x into 6 and -6 respectively by multiplying the first equation by 3 and the second by -2 to get 

6x + 9y = 15
-6x -8y = -12

Now we can combine the equations with addition and the x's will cancel leaving a single variable (which we can solve)

y = 3

Now plug this back into either original equation to find the x coordinate.

2x + 3(3) = 5

2x + 9 = 5

2x = -4

x = -2.

So your intersection point is (-2,3)