Examples
A=[312] , x=[x3yz]
Step 1
Write as a linear system of equations.
3=x
1=3y
2=z
Step 2
Step 2.1
Move variables to the left and constant terms to the right.
Step 2.1.1
Subtract x from both sides of the equation.
3−x=0
1=3y
2=z
Step 2.1.2
Subtract 3 from both sides of the equation.
−x=−3
1=3y
2=z
Step 2.1.3
Subtract 3y from both sides of the equation.
−x=−3
1−3y=0
2=z
Step 2.1.4
Subtract 1 from both sides of the equation.
−x=−3
−3y=−1
2=z
Step 2.1.5
Subtract z from both sides of the equation.
−x=−3
−3y=−1
2−z=0
Step 2.1.6
Subtract 2 from both sides of the equation.
−x=−3
−3y=−1
−z=−2
−x=−3
−3y=−1
−z=−2
Step 2.2
Write the system as a matrix.
⎡⎢
⎢⎣−100−30−30−100−1−2⎤⎥
⎥⎦
Step 2.3
Find the reduced row echelon form.
Step 2.3.1
Multiply each element of R1 by −1 to make the entry at 1,1 a 1.
Step 2.3.1.1
Multiply each element of R1 by −1 to make the entry at 1,1 a 1.
⎡⎢
⎢⎣−−1−0−0−−30−30−100−1−2⎤⎥
⎥⎦
Step 2.3.1.2
Simplify R1.
⎡⎢
⎢⎣10030−30−100−1−2⎤⎥
⎥⎦
⎡⎢
⎢⎣10030−30−100−1−2⎤⎥
⎥⎦
Step 2.3.2
Multiply each element of R2 by −13 to make the entry at 2,2 a 1.
Step 2.3.2.1
Multiply each element of R2 by −13 to make the entry at 2,2 a 1.
⎡⎢
⎢⎣1003−13⋅0−13⋅−3−13⋅0−13⋅−100−1−2⎤⎥
⎥⎦
Step 2.3.2.2
Simplify R2.
⎡⎢
⎢⎣10030101300−1−2⎤⎥
⎥⎦
⎡⎢
⎢⎣10030101300−1−2⎤⎥
⎥⎦
Step 2.3.3
Multiply each element of R3 by −1 to make the entry at 3,3 a 1.
Step 2.3.3.1
Multiply each element of R3 by −1 to make the entry at 3,3 a 1.
⎡⎢
⎢⎣100301013−0−0−−1−−2⎤⎥
⎥⎦
Step 2.3.3.2
Simplify R3.
⎡⎢
⎢⎣1003010130012⎤⎥
⎥⎦
⎡⎢
⎢⎣1003010130012⎤⎥
⎥⎦
⎡⎢
⎢⎣1003010130012⎤⎥
⎥⎦
Step 2.4
Use the result matrix to declare the final solution to the system of equations.
x=3
y=13
z=2
Step 2.5
Write a solution vector by solving in terms of the free variables in each row.
⎡⎢⎣xyz⎤⎥⎦=⎡⎢
⎢⎣3132⎤⎥
⎥⎦
Step 2.6
Write as a solution set.
⎧⎪
⎪⎨⎪
⎪⎩⎡⎢
⎢⎣3132⎤⎥
⎥⎦⎫⎪
⎪⎬⎪
⎪⎭
⎧⎪
⎪⎨⎪
⎪⎩⎡⎢
⎢⎣3132⎤⎥
⎥⎦⎫⎪
⎪⎬⎪
⎪⎭
