Solve for m1 m_1u_1+m_2u_2=m_1v_1+m_2v_2

$m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}$

Subtract $m_{1} v_{1}$ from both sides of the equation.

$m_{1} u_{1} + m_{2} u_{2} - m_{1} v_{1} = m_{2} v_{2}$

Subtract $m_{2} u_{2}$ from both sides of the equation.

$m_{1} u_{1} - m_{1} v_{1} = m_{2} v_{2} - m_{2} u_{2}$

Factor $m_{1}$ out of $m_{1} u_{1} - m_{1} v_{1}$ .

Factor $m_{1}$ out of $m_{1} u_{1}$ .

$m_{1} (u_{1}) - m_{1} v_{1} = m_{2} v_{2} - m_{2} u_{2}$

Factor $m_{1}$ out of $- m_{1} v_{1}$ .

$m_{1} (u_{1}) + m_{1} (- 1 v_{1}) = m_{2} v_{2} - m_{2} u_{2}$

Factor $m_{1}$ out of $m_{1} (u_{1}) + m_{1} (- 1 v_{1})$ .

$m_{1} (u_{1} - 1 v_{1}) = m_{2} v_{2} - m_{2} u_{2}$

Rewrite $- 1 v_{1}$ as $- v_{1}$ .

$m_{1} (u_{1} - v_{1}) = m_{2} v_{2} - m_{2} u_{2}$

Divide each term in $m_{1} (u_{1} - v_{1}) = m_{2} v_{2} - m_{2} u_{2}$ by $u_{1} - v_{1}$ and simplify.

Divide each term in $m_{1} (u_{1} - v_{1}) = m_{2} v_{2} - m_{2} u_{2}$ by $u_{1} - v_{1}$ .

$\frac{m_{1} (u_{1} - v_{1})}{u_{1} - v_{1}} = \frac{m_{2} v_{2}}{u_{1} - v_{1}} + \frac{- m_{2} u_{2}}{u_{1} - v_{1}}$

Simplify the left side.

Cancel the common factor of $u_{1} - v_{1}$ .

Cancel the common factor.

$\frac{m_{1} (u_{1} - v_{1})}{u_{1} - v_{1}} = \frac{m_{2} v_{2}}{u_{1} - v_{1}} + \frac{- m_{2} u_{2}}{u_{1} - v_{1}}$

Divide $m_{1}$ by $1$ .

$m_{1} = \frac{m_{2} v_{2}}{u_{1} - v_{1}} + \frac{- m_{2} u_{2}}{u_{1} - v_{1}}$

Simplify the right side.

Combine the numerators over the common denominator.

$m_{1} = \frac{m_{2} v_{2} - m_{2} u_{2}}{u_{1} - v_{1}}$

Simplify the numerator.

Factor $m_{2}$ out of $m_{2} v_{2} - m_{2} u_{2}$ .

Factor $m_{2}$ out of $m_{2} v_{2}$ .

$m_{1} = \frac{m_{2} (v_{2}) - m_{2} u_{2}}{u_{1} - v_{1}}$

Factor $m_{2}$ out of $- m_{2} u_{2}$ .

$m_{1} = \frac{m_{2} (v_{2}) + m_{2} (- 1 u_{2})}{u_{1} - v_{1}}$

Factor $m_{2}$ out of $m_{2} (v_{2}) + m_{2} (- 1 u_{2})$ .

$m_{1} = \frac{m_{2} (v_{2} - 1 u_{2})}{u_{1} - v_{1}}$

Rewrite $- 1 u_{2}$ as $- u_{2}$ .

$m_{1} = \frac{m_{2} (v_{2} - u_{2})}{u_{1} - v_{1}}$