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3 - simultaneous linear equations Calculator
X +
Y +
Z
=
Equation 1
X +
Y +
Z
=
Equation 2
X +
Y +
Z
=
Equation 3
Answer, X =
Answer, Y =
Answer, Z =
Result of Pivot 1, eliminating 2 and -1 in column 1
Using row 1 or equation 1 as your pivot to eliminate
and
in column 1. How do you eliminate 3?
Answer, multiply your pivot 1 equation by 3. Then subtract the product to equation 2 or row 2.
Next eliminate 2.
How ? Answer, multiply your pivot 1 equation by 2. Then subtract the product to equation 3 or row 3. The equivalent equations after eliminating 3 and 2 in column 1 is shown below.
Next you are going to use equation 2 or row 2 as your pivot 2 equation to eliminate -1 in column 2.
X + Y + Z = Equation 1
X + Y + Z = Equation 2
X + Y + Z = Equation 3
Result of Pivot 2, eliminating -1 in column 2
Using row 2 or equation 2 as your pivot to eliminate
in column 2. How do you eliminate -1 ? Answer, remember the definition of row echelon, The first nonzero entry in 2nd row must be 1. Checking your first nonzero entry in 2nd row it is not 1 instead -7. Therefore -7 is a violation of row echelon rules and it must be converted to 1. How ? Answer, by multiplying the whole equation 2 with (-1/7). Then you can use your conformed equation 2 as your pivot 2 equation to eliminate -1 in column 2. How ? By adding your pivot 2 equation to equation 3.
The equivalent equations after eliminating -1 in column 2 is shown below. You see 1 Z = 4. Using back substitution , substitute Z = 4 in equation 2, you will get Y = -2. Finally substitute Z =4, Y = -2 in equation 1, you will get X = 3.
X + Y + Z = Equation 1
X + Y + Z = Equation 2
X + Y + Z = Equation 3
Important definition to remember about row echelon. The process of reducing augmented matrix in row echelon form is called Gaussian elimination
1. The first nonzero entry in each row is always 1
2. A row with more leading zero entries compare to the previous row must be located below.
3. A row with all zero entries must be below the rows having nonzero entries.
Gaussian elimination will not work properly if one of the definition is violated.
Operation 1 - The order in which any two equations are written may be interchanged.
Operation 2 - Both sides of the equation may be multiplied by the same nonzero real number.
Operation 3 - A multiple of one equation may be added to another equation.
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