Question

Solve the system using row reduction on a calculator: 7x − 3y 2z = 19 6x 5z = 32 5x − 2y 6z = 32

Answer 1

Therefore, the solution to the given system of equations is x = 3, y = -2, and z = 4.

Given system of equations:

7x - 3y - 2z = 19

6x + 0y + 5z = 32

5x - 2y - 6z = 32

Setting up the augmented matrix:

`\[\left[\begin{array}c7 & -3 & -2 & 19 \\ 6 & 0 & 5 & 32 \\ 5 & -2 & -6 & 32\end{array}\right]\]`

Now, performing row operations to obtain the reduced row-echelon form (RREF):

Step 1: Row 2 (R2) = R2 - (6/7) × R1

Step 2: Row 3 (R3) = R3 - (5/7) × R1

Step 3: Row 3 (R3) = R3 + (2/3) × R2

This process yields the RREF matrix:

`\[\left[\begin{array}ccc1 & 0 & 0 & 3 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 4\end{array}\right]\]`

This RREF matrix represents the following system of equations:

x = 3

y = -2

z = 4