Solve {l}{frac{x_{1}}{4}+frac{x_{1}-3}{2}-1/2x_{1}=0}{frac{x_{3}-3}{2}+frac{x_{3}}{4}+1/2x_{1}=0}

$$\left\{ \begin{array} { l } { \frac { x _ { 1 } } { 4 } + \frac { x _ { 1 } - 3 } { 2 } - \frac { 1 } { 2 } x _ { 1 } = 0 } \\ { \frac { x _ { 3 } - 3 } { 2 } + \frac { x _ { 3 } } { 4 } + \frac { 1 } { 2 } x _ { 1 } = 0 } \end{array} \right.$$

Solve for x_1, x_3

$$x_{1}=6$$

$$x_{3}=-2$$

Solution Steps

Consider the first equation. Multiply both sides of the equation by 4, the least common multiple of 4,2.

$$x_{1}+2\left(x_{1}-3\right)-2x_{1}=0$$

Use the distributive property to multiply 2 by x_{1}-3.

$$x_{1}+2x_{1}-6-2x_{1}=0$$

Combine x_{1} and 2x_{1} to get 3x_{1}.

$$3x_{1}-6-2x_{1}=0$$

Combine 3x_{1} and -2x_{1} to get x_{1}.

$$x_{1}-6=0$$

Add 6 to both sides. Anything plus zero gives itself.

$$x_{1}=6$$

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

$$\frac{x_{3}-3}{2}+\frac{x_{3}}{4}+\frac{1}{2}\times 6=0$$

Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$2\left(x_{3}-3\right)+x_{3}+2\times 6=0$$

Use the distributive property to multiply 2 by x_{3}-3.

$$2x_{3}-6+x_{3}+2\times 6=0$$

Combine 2x_{3} and x_{3} to get 3x_{3}.

$$3x_{3}-6+2\times 6=0$$

Multiply 2 and 6 to get 12.

$$3x_{3}-6+12=0$$

Add -6 and 12 to get 6.

$$3x_{3}+6=0$$

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

$$3x_{3}=-6$$

Divide both sides by 3.

$$x_{3}=\frac{-6}{3}$$

Divide -6 by 3 to get -2.

$$x_{3}=-2$$

The system is now solved.

$$x_{1}=6 x_{3}=-2$$