Solve {l}{a+b+c=75}{2a-b-c=0}{30a+20b+16c=1570}

$$\left\{ \begin{array} { l } { a + b + c = 75 } \\ { 2 a - b - c = 0 } \\ { 30 a + 20 b + 16 c = 1570 } \end{array} \right.$$

Solve for a, b, c

$$a=25$$

$$b=5$$

$$c=45$$

Short Steps Using Substitution

Solve a+b+c=75 for a.

$$a=-b-c+75$$

Substitute -b-c+75 for a in the second and third equation.

$$2\left(-b-c+75\right)-b-c=0 30\left(-b-c+75\right)+20b+16c=1570$$

Solve these equations for b and c respectively.

$$b=-c+50 c=\frac{340}{7}-\frac{5}{7}b$$

Substitute -c+50 for b in the equation c=\frac{340}{7}-\frac{5}{7}b.

$$c=\frac{340}{7}-\frac{5}{7}\left(-c+50\right)$$

Solve c=\frac{340}{7}-\frac{5}{7}\left(-c+50\right) for c.

$$c=45$$

Substitute 45 for c in the equation b=-c+50.

$$b=-45+50$$

Calculate b from b=-45+50.

$$b=5$$

Substitute 5 for b and 45 for c in the equation a=-b-c+75.

$$a=-5-45+75$$

Calculate a from a=-5-45+75.

$$a=25$$

The system is now solved.

$$a=25 b=5 c=45$$