2D collision of billiard balls

[PStudent111]

Homework Statement

Two billiard balls of equal mass move at right angles and meet at the origin of an xy coordinate system. Initially ball A is moving along the y axis at +2.0 m/s, and ball B is moving to the right along the x axis with speed +3.7 m/s. After the collision (assumed elastic), the second ball is moving along the positive y axis.

What is the final direction of ball A, and what are the speeds of the two balls?

Homework Equations

x direction: m_bv_b=m_av_a^'cosθ_a

y direction: m_av_a=m_av_a^'sinθ_a+m_bv_b^'sinθ_b

conservation of KE: v_a²+v_b²=v_a^'2+v_b^'2

Alright, so I've played around with the equations and I just cannot seem to arrive at the solution by myself. I do have the solutions manual and I understand the way in which they arrived at the solution, but I honestly would never have thought to manipulate the equations in the way they did. So, I'm posting the problem on here to see if you guys have an alternative way to solve these types of problems.

 

[voko]

The second ball has velocity only in the Y direction, the first ball may still have velocity components in both X and Y directions. That gives you three unknowns. You have conservation of momentum in X and Y - two equations - and conservation of KE because the collision is elastic, thus three equations for three variables. Solve the system.