Solve 2a + b - c = - 1; a + 3b - 2c = - 6; 5a + 2b + c = 5 | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$2a+b-c=-1; a+3b-2c=-6; 5a+2b+c=5$$

Solve for a, b, c

$a=1$
$b=-1$
$c=2$

Short Steps Using Substitution

Solve $2a+b-c=-1$ for $b$.

$$b=-2a+c-1$$

Substitute $-2a+c-1$ for $b$ in the second and third equation.

$$a+3\left(-2a+c-1\right)-2c=-6$$ $$5a+2\left(-2a+c-1\right)+c=5$$

Solve these equations for $a$ and $c$ respectively.

$$a=\frac{3}{5}+\frac{1}{5}c$$ $$c=-\frac{1}{3}a+\frac{7}{3}$$

Substitute $\frac{3}{5}+\frac{1}{5}c$ for $a$ in the equation $c=-\frac{1}{3}a+\frac{7}{3}$.

$$c=-\frac{1}{3}\left(\frac{3}{5}+\frac{1}{5}c\right)+\frac{7}{3}$$

Solve $c=-\frac{1}{3}\left(\frac{3}{5}+\frac{1}{5}c\right)+\frac{7}{3}$ for $c$.

$$c=2$$

Substitute $2$ for $c$ in the equation $a=\frac{3}{5}+\frac{1}{5}c$.

$$a=\frac{3}{5}+\frac{1}{5}\times 2$$

Calculate $a$ from $a=\frac{3}{5}+\frac{1}{5}\times 2$.

$$a=1$$

Substitute $1$ for $a$ and $2$ for $c$ in the equation $b=-2a+c-1$.

$$b=-2+2-1$$

Calculate $b$ from $b=-2+2-1$.

$$b=-1$$

The system is now solved.

$$a=1$$ $$b=-1$$ $$c=2$$