Solve {c}{A+B=0}{-4A-B+C=4}{4A-2B+C=5}

$$\left\{ \begin{array} { c } { A + B = 0 } \\ { - 4 A - B + C = 4 } \\ { 4 A - 2 B + C = 5 } \end{array} \right.$$

Solve for A, B, C

$$A=\frac{1}{9}\approx 0.111111111$$

$$B=-\frac{1}{9}\approx -0.111111111$$

$$C = \frac{13}{3} = 4\frac{1}{3} \approx 4.333333333$$

Short Steps Using Substitution

Solve A+B=0 for A.

$$A=-B$$

Substitute -B for A in the second and third equation.

$$-4\left(-1\right)B-B+C=4 4\left(-1\right)B-2B+C=5$$

Solve these equations for B and C respectively.

$$B=\frac{4}{3}-\frac{1}{3}C C=6B+5$$

Substitute \frac{4}{3}-\frac{1}{3}C for B in the equation C=6B+5.

$$C=6\left(\frac{4}{3}-\frac{1}{3}C\right)+5$$

Solve C=6\left(\frac{4}{3}-\frac{1}{3}C\right)+5 for C.

$$C=\frac{13}{3}$$

Substitute \frac{13}{3} for C in the equation B=\frac{4}{3}-\frac{1}{3}C.

$$B=\frac{4}{3}-\frac{1}{3}\times \frac{13}{3}$$

Calculate B from B=\frac{4}{3}-\frac{1}{3}\times \frac{13}{3}.

$$B=-\frac{1}{9}$$

Substitute -\frac{1}{9} for B in the equation A=-B.

$$A=-\left(-\frac{1}{9}\right)$$

Calculate A from A=-\left(-\frac{1}{9}\right).

$$A=\frac{1}{9}$$

The system is now solved.

$$A=\frac{1}{9} B=-\frac{1}{9} C=\frac{13}{3}$$