Question

How do you solve the system $2x+y-z=2$, $-x-3y+z=-1$ and $-4x+3y+z=-4$?

Answer 1

In this problem, we have to solve the given system of equations to find the value of x, y and z. We can first take the equation (1) and (2) for subtraction to get the equation (4), then we can take the equation (2) and (3) for subtraction to get the new equation (6), we can the solve the equation (4) and (6), to get any one of the values of x, y, z. We can substitute the resulting values in any of the equations to get another value. We can repeat this method until finding all the three values.

Complete step by step solution:
We know that the given system of equations to be solved are,
$2x+y-z=2$ ……… (1)
$-x-3y+z=-1$ …….. (2)
$-4x+3y+z=-4$……. (3)
We can now subtract the equation by elimination method.
We should know that to solve by elimination method, we should have similar terms to be cancelled, so we can multiply both equations with numbers to get similar terms.
We can now add the equation (1) and (2), we get

& \Rightarrow 2x+y-z-2-x-3y+z=0 \\\ & \Rightarrow x-2y-1=0.....(4) \\\ \end{aligned}$$ We can now subtract the equation (2) and (3), we get $$\begin{aligned} & \Rightarrow -x-3y+z+1-\left( -4x+3y+z+4 \right)=0 \\\ & \Rightarrow -3x+6y=-3......(5) \\\ \end{aligned}$$ Now we can multiply 3 in equation (4), we get $$\Rightarrow 3x-6y=3$$ …… (6) Now we can add the above equations (5) and (6), we get $$\Rightarrow -3x+6y+3+3x-6y-3=0$$ Now we can cancel similar terms and simplify, we get $$\Rightarrow 0=0$$ Similarly, if we do elimination method for equation (2) and (3), we will get the same result, $$\Rightarrow 0=0$$ **Therefore, the given three systems of equations have infinitely many solutions.** **Note:** Students may get confused in case of three equations given to find three unknown variables. We can take two equations first to solve for one new equation and the next two equations, next to solve for another equation, then we can solve those new equations to get any one of the values. We can substitute that value in any equation to get the other value respectively.