Question

Solve the following sets of simultaneous equations by reducing the matrix to row echelon form. $$ \left\\{\begin{array}{c} 2 x+3 z=5 \\ 4 x+2 y+5 z=13 \\ x-y+2 z=1 \end{array}\right. $$

Short answer

x = 2, y = 1, z = 1

Step-by-step solution

- Write the augmented matrix

First, write the system of equations as an augmented matrix: ooo

- Convert first element in first column to 1

Divide the first row by 2 to make the first element in the first row 1. ooo

- Eliminate the first column elements below the first row

Subtract twice the first row from the second row, and add the first row to the third row to get zeros below the pivot element in the first column. ooo

- Convert the leading element of second row to 1

Divide the second row by -4 to make the leading element in the second row 1. ooo

- Eliminate elements below the leading element of the second row

Add twice the second row to the third row to get zeros below the pivot element in the second column. ooo

- Back-substitution to find solution

Use back-substitution to find the values of each variable. Start from the last row and substitute back to find values of z, y and x.oo

Key concepts

augmented matrix

In solving simultaneous equations, we often start by writing them as an augmented matrix. An augmented matrix is a compact way to include the coefficients of the variables and the constants from the right-hand side of the equations.
It helps in simplifying the process of Cartesian calculations. For example, the equations given: \[ \left\{ \begin{array}{c} 2 x + 3 z = 5 \ 4 x + 2 y + 5 z = 13 \ x - y + 2 z = 1 \end{array}\right. \] can be represented as the augmented matrix: \[ \left[ \begin{array}{ccc|c} 2 & 0 & 3 & 5 \ 4 & 2 & 5 & 13 \ 1 & -1 & 2 & 1 \end{array}\right] \].

Gaussian elimination

Gaussian elimination is a method for solving systems of linear equations. It involves three main steps: forward elimination, achieving row echelon form, and back-substitution. The goal is to use row operations to transform the matrix into a simpler form.
These operations are:

• Swapping rows
• Multiplying a row by a non-zero constant
• Adding or subtracting a multiple of one row to another

Applying these steps systematically brings the matrix to row echelon form, which looks like a staircase.

back-substitution

Back-substitution is used once the augmented matrix is in row echelon form. Starting from the bottom row, solve for one variable at a time.
For example, in a system where the matrix is: \[ \left[ \begin{array}{ccc|c} 1 & a_{12} & a_{13} & b_1 \ 0 & 1 & a_{23} & b_2 \ 0 & 0 & 1 & b_3 \end{array}\right] \] you would first find the value of the variable associated with the bottom row, then substitute that value into the rows above to find the other variables.
This process continues until all variables are solved.

simultaneous equations

Simultaneous equations are a set of equations with multiple variables that are solved together. Each equation in the set provides additional constraints.
The solution is the set of variable values that satisfy all equations simultaneously. For example, consider the set: \[ \left\{ \begin{array}{c} 2 x + 3 z = 5 \ 4 x + 2 y + 5 z = 13 \ x - y + 2 z = 1 \end{array}\right. \]
The aim is to find values for x, y, and z that make all three equations true at the same time. Using methods like Gaussian elimination and back-substitution is essential in finding these solutions.