Solve {l}{x+y=a}{2x+4z=1}{5y-2z=3}

$$\left. \begin{array} { l } { x + y = a } \\ { 2 x + 4 z = 1 } \\ { 5 y - 2 z = 3 } \end{array} \right.$$

Solve for x, y, z

$$x=\frac{5a}{4}-\frac{7}{8}$$

$$y=-\frac{a}{4}+\frac{7}{8}$$

$$z=-\frac{5a}{8}+\frac{11}{16}$$

Short Steps Using Substitution

Solve x+y=a for x.

$$x=-y+a$$

Substitute -y+a for x in the equation 2x+4z=1.

$$2\left(-y+a\right)+4z=1$$

Solve the second equation for y and the third equation for z.

$$y=a+2z-\frac{1}{2} z=\frac{5}{2}y-\frac{3}{2}$$

Substitute a+2z-\frac{1}{2} for y in the equation z=\frac{5}{2}y-\frac{3}{2}.

$$z=\frac{5}{2}\left(a+2z-\frac{1}{2}\right)-\frac{3}{2}$$

Solve z=\frac{5}{2}\left(a+2z-\frac{1}{2}\right)-\frac{3}{2} for z.

$$z=\frac{11}{16}-\frac{5}{8}a$$

Substitute \frac{11}{16}-\frac{5}{8}a for z in the equation y=a+2z-\frac{1}{2}.

$$y=a+2\left(\frac{11}{16}-\frac{5}{8}a\right)-\frac{1}{2}$$

Calculate y from y=a+2\left(\frac{11}{16}-\frac{5}{8}a\right)-\frac{1}{2}.

$$y=\frac{7}{8}-\frac{1}{4}a$$

Substitute \frac{7}{8}-\frac{1}{4}a for y in the equation x=-y+a.

$$x=-\left(\frac{7}{8}-\frac{1}{4}a\right)+a$$

Calculate x from x=-\left(\frac{7}{8}-\frac{1}{4}a\right)+a.

$$x=-\frac{7}{8}+\frac{5}{4}a$$

The system is now solved.

$$x=-\frac{7}{8}+\frac{5}{4}a y=\frac{7}{8}-\frac{1}{4}a z=\frac{11}{16}-\frac{5}{8}a$$