Question

This is my first time doing linear equations in four variables and my answers came out to be very odd numbers which leads me to believe that they are incorrect, but I am not sure. How would you solve this problem? I have to eliminate X first.

Answer 1

When we have four equations, we have to eliminate variables so we form 2x2 systems, that is, two equations with two variables.

First, multiply the first equation by -1, to then combine it with the second equation.

`\begin{gathered} -x-y+z-w+x+3y-4z+5w=-10+26 \\ 2y-3z+4w=16 \end{gathered}`

Let's repeat the process with equations 3 and 4.

`\begin{gathered} -x-4y+7z-7w+x+2y-3z+6w=-37+22 \\ -2y+4z-w=-15 \end{gathered}`

Repeat the process one more time because we need three equations with the variables y, z, and w. This time let's do it with equations 1 and 3.

`\begin{gathered} -x-y+z-w+x+4y-7z+7w=-10+37 \\ 3y+6w-6z=27 \\ y+2w-2z=9 \end{gathered}`

Now, combine the first two equations we've got.

`\begin{gathered} 2y-3z+4w-2y+4z-w=16-15 \\ z+3w=1 \end{gathered}`

We need another equation with the variables z and w only. To find it, multiply the third equation by 2 and combine it with equation number 2 above.

`\begin{gathered} 2y+4w-4z-2y+4z-w=18-15 \\ 3w=3 \\ w=1 \end{gathered}`

The value of w is 1.

Let's use the w-value in the equation z+3w=1 to find z.

`\begin{gathered} z+3(1)=1 \\ z+3=1 \\ z=1-3 \\ z=-2 \end{gathered}`

The value of z is -2.

Now we can find the value of y using the equation y+2w-2z=9.

`\begin{gathered} y+2(1)-2(-2)=9 \\ y+2+4=9 \\ y+6=9 \\ y=9-6 \\ y=3 \end{gathered}`

The value of y is 3.

At last, we can find x using any equation because we already know three variables.

`\begin{gathered} x+y-z+w=10 \\ x+3-(-2)+1=10 \\ x+3+2+1=10 \\ x+6=10 \\ x=10-6 \\ x=4 \end{gathered}`

The value of x is 4.

Therefore, the solutions to the system are x = 4, y = 3, z = -2, and w = 1.