Chapter 10, Problem 37RE

To determine

To find: the solution of the system.

Answer to Problem 37RE

The solution is $\left(s+1,2s-t+1,s,t\right)$

Explanation of Solution

Given:

  $\{\begin{array}{l}x-y+z-w=0\\ 3x-y-z-w=2\end{array}$

Calculation:

Let's begin by labeling our two equations.

  $\{\begin{array}{l}x-y+z-w=0 Equation1\\ 3x-y-z-w=2 Equation\text{ 2}\end{array}$

Use Operation $1$ to subtract Equation $1$ from Equation $\text{2}$ .

  $\begin{array}{l}3x-y-z-w=2 Equation\text{ 2}\\ +\left[x-y+z-w=0\right] Equation\text{ 1}\\ 2x-2z=2 Equation\text{ 2} - Equation\text{ 1}\end{array}$

This reduces to

  $x-z=1$

Solving for $x$ ,

  $x=z+1$

This is a true relationship between the variables $x$ and $z$ .

Now, let's add the original two equations to get another true statement.

  $\begin{array}{l}3x-y-z-w=2 Equation\text{ 2}\\ +\left[x-y+z-w=0\right] Equation\text{ 1}\\ 4x-2y-2w=2 Equation\text{ 2} - Equation\text{ 1}\end{array}$

This reduces to

  $2x-y-w=1$

Divide both sides by $2$ . Insert our value of $x=z+1$ into this equation.

  $\begin{array}{l}2\left(z+1\right)-y-w=1\\ 2x+2-y-w=1\\ -y+2z-w+2=1\end{array}$

Solve for $y$ .

  $y=2z-w+1$

Now $xandy$ are in terms of $zandw$ . Choose $zandw$ to be the independent variables $sandt$ (respectively). Let $sandt$ represent any real numbers. Since $sandt$ can take on any real values, our system has infinitely many solutions. The complete solution to the system is

  $\begin{array}{l}x=s+1\\ y=2s-t+1\\ z=s\\ w=t\end{array}$

This solution can also be written as the ordered quadruplet

  $\left(s+1,2s-t+1,s,t\right)$

This solution represents a plane. For every value of the ordered pair $\left(s,t\right),$ there is a solution at the point $\left(s+1,2s-t+1,s,t\right)$ .

Conclusion: There is a solution at the point $\left(s+1,2s-t+1,s,t\right)$