Solve Using a Matrix by Elimination w+z=400 , w+x=630 , x-y=180 , y-z=50

$w + z = 400$ , $w + x = 630$ , $x - y = 180$ , $y - z = 50$

Step 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 0 & 0 & 1 & 400 \\ 1 & 1 & 0 & 0 & 630 \\ 0 & 1 & - 1 & 0 & 180 \\ 0 & 0 & 1 & - 1 & 50 \end{array}]$

Step 2

Find the reduced row echelon form.

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 0 & 0 & 1 & 400 \\ 1 - 1 & 1 - 0 & 0 - 0 & 0 - 1 & 630 - 400 \\ 0 & 1 & - 1 & 0 & 180 \\ 0 & 0 & 1 & - 1 & 50 \end{array}]$

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 0 & 1 & 400 \\ 0 & 1 & 0 & - 1 & 230 \\ 0 & 1 & - 1 & 0 & 180 \\ 0 & 0 & 1 & - 1 & 50 \end{array}]$

Perform the row operation $R_{3} = R_{3} - R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 0 & 0 & 1 & 400 \\ 0 & 1 & 0 & - 1 & 230 \\ 0 - 0 & 1 - 1 & - 1 - 0 & 0 + 1 & 180 - 230 \\ 0 & 0 & 1 & - 1 & 50 \end{array}]$

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 0 & 1 & 400 \\ 0 & 1 & 0 & - 1 & 230 \\ 0 & 0 & - 1 & 1 & - 50 \\ 0 & 0 & 1 & - 1 & 50 \end{array}]$

Multiply each element of $R_{3}$ by $- 1$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & 0 & 0 & 1 & 400 \\ 0 & 1 & 0 & - 1 & 230 \\ - 0 & - 0 & - - 1 & - 1 \cdot 1 & - - 50 \\ 0 & 0 & 1 & - 1 & 50 \end{array}]$

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 0 & 1 & 400 \\ 0 & 1 & 0 & - 1 & 230 \\ 0 & 0 & 1 & - 1 & 50 \\ 0 & 0 & 1 & - 1 & 50 \end{array}]$

Perform the row operation $R_{4} = R_{4} - R_{3}$ to make the entry at $4, 3$ a $0$ .

$[\begin{array}{c} 1 & 0 & 0 & 1 & 400 \\ 0 & 1 & 0 & - 1 & 230 \\ 0 & 0 & 1 & - 1 & 50 \\ 0 - 0 & 0 - 0 & 1 - 1 & - 1 + 1 & 50 - 50 \end{array}]$

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & 0 & 0 & 1 & 400 \\ 0 & 1 & 0 & - 1 & 230 \\ 0 & 0 & 1 & - 1 & 50 \\ 0 & 0 & 0 & 0 & 0 \end{array}]$

Step 3

Use the result matrix to declare the final solution to the system of equations.

$w + z = 400 x - z = 230 y - z = 50 0 = 0$

Step 4

The solution is the set of ordered pairs that make the system true.

$(400 - z, 230 + z, 50 + z, z)$