Solve {l}{(u-v/2)+3=u-v/2}{u+3v/2=2-u-v/2}

$$\left. \begin{array}{l}{ ( u - \frac { v } { 2 } ) + 3 = \frac { u - v } { 2 } }\\{ \frac { u + 3 v } { 2 } = 2 - \frac { u - v } { 2 } }\end{array} \right.$$

Solve for u, v

$$u=-6$$

$$v=8$$

Solution Steps

Consider the first equation. Multiply both sides of the equation by 2.

$$2u-v+6=u-v$$

Subtract u from both sides.

$$2u-v+6-u=-v$$

Combine 2u and -u to get u.

$$u-v+6=-v$$

Add v to both sides.

$$u-v+6+v=0$$

Subtract 6 from both sides. Anything subtracted from zero gives its negation.

$$u-v+v=-6$$

Combine -v and v to get 0.

$$u=-6$$

Consider the second equation. Insert the known values of variables into the equation.

$$\frac{-6+3v}{2}=2-\frac{-6-v}{2}$$

Multiply both sides of the equation by 2.

$$-6+3v=4-\left(-6-v\right)$$

To find the opposite of -6-v, find the opposite of each term.

$$-6+3v=4+6+v$$

Add 4 and 6 to get 10.

$$-6+3v=10+v$$

Subtract v from both sides.

$$-6+3v-v=10$$

Combine 3v and -v to get 2v.

$$-6+2v=10$$

Add 6 to both sides.

$$2v=10+6$$

Add 10 and 6 to get 16.

$$2v=16$$

Divide both sides by 2.

$$v=\frac{16}{2}$$

Divide 16 by 2 to get 8.

$$v=8$$

The system is now solved.

$$u=-6 v=8$$