Solve {l}{4x-2y=6}{3x+7y=10}{z=x+y}

$$\left. \begin{array} { l } { 4 x - 2 y = 6 } \\ { 3 x + 7 y = 10 } \\ { z = x + y } \end{array} \right.$$

Solve for x, y, z

$$x = \frac{31}{17} = 1\frac{14}{17} \approx 1.823529412$$

$$y=\frac{11}{17}\approx 0.647058824$$

$$z = \frac{42}{17} = 2\frac{8}{17} \approx 2.470588235$$

Short Steps Using Substitution

Reorder the equations.

$$z=x+y 3x+7y=10 4x-2y=6$$

Solve z=x+y for x.

$$x=z-y$$

Substitute z-y for x in the second and third equation.

$$3\left(z-y\right)+7y=10 4\left(z-y\right)-2y=6$$

Solve these equations for y and z respectively.

$$y=-\frac{3}{4}z+\frac{5}{2} z=\frac{3}{2}+\frac{3}{2}y$$

Substitute -\frac{3}{4}z+\frac{5}{2} for y in the equation z=\frac{3}{2}+\frac{3}{2}y.

$$z=\frac{3}{2}+\frac{3}{2}\left(-\frac{3}{4}z+\frac{5}{2}\right)$$

Solve z=\frac{3}{2}+\frac{3}{2}\left(-\frac{3}{4}z+\frac{5}{2}\right) for z.

$$z=\frac{42}{17}$$

Substitute \frac{42}{17} for z in the equation y=-\frac{3}{4}z+\frac{5}{2}.

$$y=-\frac{3}{4}\times \frac{42}{17}+\frac{5}{2}$$

Calculate y from y=-\frac{3}{4}\times \frac{42}{17}+\frac{5}{2}.

$$y=\frac{11}{17}$$

Substitute \frac{11}{17} for y and \frac{42}{17} for z in the equation x=z-y.

$$x=\frac{42}{17}-\frac{11}{17}$$

Calculate x from x=\frac{42}{17}-\frac{11}{17}.

$$x=\frac{31}{17}$$

The system is now solved.

$$x=\frac{31}{17} y=\frac{11}{17} z=\frac{42}{17}$$