Question

How do you solve linear equations in three variables?

Answer 1

Linear equations in three variable:

Step 1. Take any two equation out given three equation, and solve it for one variable.

Again take two equation and solve it for same variable as above.

Now solve the two equation so form. and find their value, and put the value in any of the three equation.

For example,

$\begin{array}{rcl}x-2y+3z& =& 9......\left(1\right)\\ -x+3y-z& =& -6\dots \dots \left(2\right)\\ 2x-5y+5z& =& 17.\dots \dots \left(3\right)\end{array}$

Step 2. Add equation (1) and (2) to eliminate $x$.

$$\begin{gathered} x-2y+3z+(-x+3y-z)=9–(-6) \\ y+2z=15.....\left(4\right) \end{gathered}$$

Step 3. Solve, equation $1$ and $3$, by multiplying equation (1) with $-2$ and adding to equation (3)

$$\begin{gathered} -2x+4y-6z=-18+(2x-5y+5z=17) \\ =-18+1 \end{gathered}$$

$-y-z=-1\dots ..\left(5\right)$

Step 4. Solve for equation $4\&5$

$$\begin{gathered} \begin{array}{rcl}y+2z& =& 3\\ -y-z& =& -1\\ & & ----------- \\ \\ & & z=2\\ & & y=-1\end{array} \end{gathered}$$

Solve for $x$, we get $x=1$

Therefore, the solution for the given three equations is $(1,-1,2).$

Hence, in this way we can solve linear equation in three variable.