Solve {l}{x+y+z=8}{5x-y=7}{7z+9=10}

$$\left\{ \begin{array} { l } { x + y + z = 8 } \\ { 5 x - y = 7 } \\ { 7 z + 9 = 10 } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{52}{21} = 2\frac{10}{21} \approx 2.476190476$$

$$y = \frac{113}{21} = 5\frac{8}{21} \approx 5.380952381$$

$$z=\frac{1}{7}\approx 0.142857143$$

Short Steps Using Substitution

Solve x+y+z=8 for x.

$$x=-y-z+8$$

Substitute -y-z+8 for x in the equation 5x-y=7.

$$5\left(-y-z+8\right)-y=7$$

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

$$y=-\frac{5}{6}z+\frac{11}{2} z=\frac{1}{7}$$

Substitute \frac{1}{7} for z in the equation y=-\frac{5}{6}z+\frac{11}{2}.

$$y=-\frac{5}{6}\times \frac{1}{7}+\frac{11}{2}$$

Calculate y from y=-\frac{5}{6}\times \frac{1}{7}+\frac{11}{2}.

$$y=\frac{113}{21}$$

Substitute \frac{113}{21} for y and \frac{1}{7} for z in the equation x=-y-z+8.

$$x=-\frac{113}{21}-\frac{1}{7}+8$$

Calculate x from x=-\frac{113}{21}-\frac{1}{7}+8.

$$x=\frac{52}{21}$$

The system is now solved.

$$x=\frac{52}{21} y=\frac{113}{21} z=\frac{1}{7}$$