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Solving a system of linear equations

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Wize University Linear Algebra Textbook > Systems of Linear Equations (SLEs) (Linear Systems)

Reducing a Matrix (Gauss-Jordan Elimination)

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Consider the following system of linear equations:
−2x1−4x2−6x3−19x4=−153x1+10x2−30x3+9x4=24x1+3x2−6x3+4x4=87x1+17x2+45x4=55\begin{array}{rcr} -2x_1-4x_2-6x_3-19x_4&=&-15\\ 3x_1+10x_2-30x_3+9x_4&=&24\\ x_1+3x_2-6x_3+4x_4&=&8\\ 7x_1+17x_2+45x_4&=&55 \end{array}
where we wish to solve for the quadruple (x1, x2, x3, x4) of real numbers that satisfy this system.

(a) Write the augmented matrix of this linear system.

(b) Transform the augmented matrix into row echelon form using elementary row operations. Clearly indicate which operation you are performing at each step.

(c) Use your result from part (b) to find the solution set to the original system using back substitution.
More Reducing a Matrix (Gauss-Jordan Elimination) Questions:
Gauss-Jordan Elimination $\tkct{LAT}$
Gauss-Jordan Elimination $\tkct{LAT}$
Use Gauss-Jordan Elimination to find the RREF of the matrix [0−23  1363−2663  5]\bcb{\begin{bmatrix} 0&-2&3&\ \ 1\\3&6&3&-2\\ 6&6&3&\ \ 5 \end{bmatrix}}.
Given the following augmented matrix of a system of equations, find the reduced row echelon form:
[1011−23−22−4∣110]\left[\begin{array}{l} 1&0&1\\ 1&-2&3\\ -2&2&-4\\ \end{array}\right.\left|\begin{array}{l}1 \\1\\0\end{array}\right]
Solving a system of linear equations
Consider the following system of linear equations:
−2x1+3x2+13x3=5−x1+2x2+5x3=14x1−14x2−2x3=14\begin{array}{rcr} -2x_1+3x_2+13x_3&=&5\\ -x_1+2x_2+5x_3&=&1\\ 4x_1-14x_2-2x_3&=&14\\ \end{array}
where we wish to solve for the triple (x1, x2, x3) of real numbers that satisfy this system.
Additional Practice Problems--Systems of Linear Equations
1. Write the augmented matrix that represents this system of linear equations
x1+x2=7−3x32−x1=7x1+x2+x3=0\begin{array}{rcl} x_1+x_2&=&7-3x_3\\ 2-x_1&=&7\\ x_1+x_2+x_3&=&0 \end{array}
2. Determine all values of a, ba,\ b and cc if the augmented matrix [2−1001−3]\left[\begin{array}{rr|r} 2&-1&0\\ 0&1&-3 \end{array}\right] represents the following system of linear equations: ax1=bx2(a+c)x2+3=0\begin{array}{rcl} ax_1&=&bx_2\\ (a+c)x_2+3&=&0 \end{array}
Reducing a Matrix
Practice: Reducing a Matrix
Find the reduced row echelon form (RREF) of the matrix:
[312101−124]\begin{bmatrix} 3&1&2\\ 1&0&1\\ -1&2&4 \end{bmatrix}
Reducing a Matrix
Practice: Reducing a Matrix
Given the matrix,
[13−13−6−321−2]\left[\begin{array}{ccc} 1&3&-1\\ 3&-6&-3\\ 2&1&-2 \end{array}\right]
Reducing a Matrix
Given the matrix,
[13−13−6−321−2]\left[\begin{array}{ccc} 1&3&-1\\ 3&-6&-3\\ 2&1&-2 \end{array}\right]
find the first row of the RREF of this matrix.
Practice Question: Matrix Reduction
Practice Question: Matrix Reduction
Reduce [6−1812302−64024−1280−1−26400]\left[\begin{array}{rrrr|r} 6 & -18 & 12 &3 &0\\ 2 &-6 & 4 &0&2\\ 4 & -12 & 8 &0&-1\\ -2&6&4&0&0 \end{array}\right] into row-reduce echelon form (RREF) and then state the solution.
Let AA be the row-reduced echelon form of the matrix [120−20132−1]\left[\begin{array}{r} 1&2&0\\ -2&0&1\\ 3&2&-1 \end{array}\right].
Find the first row of AA.
Which of the following matrices is a result of performing a single elementary row operation on the augmented matrix [2−2051−43100−24]\left[\begin{array}{rrr|r} 2&-2&0&5\\ 1&-4&3&1\\ 0&0&-2&4 \end{array}\right]?
Solving a system of linear equations
Consider the following system of linear equations:
−2x1+3x2+13x3=5−x1+2x2+5x3=14x1−14x2−2x3=14\begin{array}{rcr} -2x_1+3x_2+13x_3&=&5\\ -x_1+2x_2+5x_3&=&1\\ 4x_1-14x_2-2x_3&=&14\\ \end{array}
where we wish to solve for the triple (x1, x2, x3) of real numbers that satisfy this system.
Solving a system of linear equations
Consider the following system of linear equations:
−2x1−4x2−6x3−19x4=−153x1+10x2−30x3+9x4=24x1+3x2−6x3+4x4=87x1+17x2+45x4=55\begin{array}{rcr} -2x_1-4x_2-6x_3-19x_4&=&-15\\ 3x_1+10x_2-30x_3+9x_4&=&24\\ x_1+3x_2-6x_3+4x_4&=&8\\ 7x_1+17x_2+45x_4&=&55 \end{array}
where we wish to solve for the quadruple (x1, x2, x3, x4) of real numbers that satisfy this system.
Solving a system of linear equations
Consider the following system of linear equations:
−2x1+3x2+13x3=5−x1+2x2+5x3=14x1−14x2−2x3=14\begin{array}{rcr} -2x_1+3x_2+13x_3&=&5\\ -x_1+2x_2+5x_3&=&1\\ 4x_1-14x_2-2x_3&=&14\\ \end{array}
where we wish to solve for the triple (x1, x2, x3) of real numbers that satisfy this system.
Solving a system of linear equations
Consider the following system of linear equations:
−2x1−4x2−6x3−19x4=−153x1+10x2−30x3+9x4=24x1+3x2−6x3+4x4=87x1+17x2+45x4=55\begin{array}{rcr} -2x_1-4x_2-6x_3-19x_4&=&-15\\ 3x_1+10x_2-30x_3+9x_4&=&24\\ x_1+3x_2-6x_3+4x_4&=&8\\ 7x_1+17x_2+45x_4&=&55 \end{array}
where we wish to solve for the quadruple (x1,x2,x3,x4)(x_1,x_2,x_3,x_4) of real numbers that satisfy this system
Solving a system of linear equations
Consider the following system of linear equations:
−2x1+3x2+13x3=5−x1+2x2+5x3=14x1−14x2−2x3=14\begin{array}{rcr} -2x_1+3x_2+13x_3&=&5\\ -x_1+2x_2+5x_3&=&1\\ 4x_1-14x_2-2x_3&=&14\\ \end{array}
where we wish to solve for the triple (x1,x2,x3)(x_1,x_2,x_3) of real numbers that satisfy this system
Augmented Matrix: Gauss-Jordan Elimination
Consider the system of linear equations
x2+4x3=14+2x1−4x4x1−x3−x4+3=02x4−6=0\begin{array}{c} x_2+4x_3=14+2x_1-4x_4\\ x_1-x_3-x_4+3=0\\ 2x_4-6=0 \end{array}
(a) Write the augmented matrix for the standard form of this system.