# How to solve the system of equations with three variables?

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**Hint**: As we know that an equation for a straight line is called a linear equation. The standard form of linear equations in three variables is $Ax + By + Cz = R$ . In this question we will take linear equations in three variables and how to solve them by elimination method.

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__Complete step-by-step answer__We can solve linear equations in three variables by elimination method. Here is step by step how we can solve equations with three variables:

First we pick any two equations from the system and then eliminate the same variable from each pair using the addition/subtraction method.

Now we solve the new two equations by the same method as above.

Then we substitute the solution back into one of the original equations and solve for the third variable.

Let us take an example of equations $4x - 3y + z = - 10,2x + y + 3z = 0$ and $ - x + 2y - 5z = 17$.

Now we take the first two equations and multiply the second equation by 2 and then subtract it:$4x - 3y + z - \{ 2(2x + y + 3z) = - 10 - 0$.

We have two new equations$4x - 3y + z - 4x - 2y - 6z = - 10$.

It gives us a new equation $ - 5y - 5z = - 10$.

Now we solve the last two equations by multiplying the third equation with $2$ and adding it:$2x + y + 3z - \{ 2( - x + 2y - 5)\} = 0 - 17 \times 2$.

On further solving by breaking the brackets we have: $2x + y + 3z - 2x + 4y - 10z = 0 - 34$, It gives us a new equation $5y - 7z = 34$.

We now have two new equations. Now we add both the equations: $ - 5y - 5z + 5y - 7z = - 10 + 34$.

On further solving we get $ - 12z = 24 \Rightarrow z = - 2$.

By putting this value in the equation: $ - 5y - 5( - 2) = - 10 \Rightarrow - 5y = - 10 - 10$.

It gives the value $y = \dfrac{{ - 20}}{{ - 5}} = 4$.

Now we substitute the value in the original equation: $ - x + 2(4) - 5( - 2) = 17$.

On further solving it gives

$ - x + 18 = 17 \Rightarrow - x = - 1$ or $x = 1$.

Hence this is the method we apply to solve the equations with three variables.

**Note**:We should note that there is an alternate way to solve linear equations in three variables i.e. Addition Method. Let us assume that the equations are in the form $x + y = a,y + z = b,z + x = c$.

We can add all the three equations and divide the right hand side by $2$ i.e. $x + y + z = \dfrac{{a + b + c}}{2}$. Now we can find each variable separately by just subtracting the original equations from the last equation. Let us say we will subtract the first given equation i.e. $x + y + z - x - y = \dfrac{{a + b + c}}{2} - a$.

On solving it gives us $z = \dfrac{{b + c + a - 2a}}{2} \Rightarrow z = \dfrac{{b + c - a}}{2}$. Similarly on subtracting it gives us $y = \dfrac{{a + b - c}}{2}$ and $x = \dfrac{{a + c - b}}{2}$ .

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