| 3.1 |
Systems of Linear Equations in Three Variables
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| \(\blacksquare\) Two or more linear equations involving the same set of variables form a system of linear equations. |
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\(\blacksquare\) The characteristics of systems of linear equations in three variables:
- Has three variables in each linear equation.
- The highest power of each variable is \(1\).
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| Example of system of linear equations in three variables: |
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\(\boxed{\begin{aligned} 4x-2y+z&=2\quad \\\\ 6x+7y-z&=3 \\\\ 5x+y+2z&=7 \end{aligned} }\)
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| \(\blacksquare\) Geometrically, a linear equation in three variables forms a plane in a three-dimensional space. |
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| \(\blacksquare\) There are three types of solutions for the systems of linear equations in three variables: |
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| One Solution |
Infinite
solutions |
No solution |
The planes
intersect at only
one point |
The planes
intersect in a
straight line |
The planes do
not intersect at
any point |
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Methods used to solve systems of linear equations in three variables:
- Elimination method
- Substitution method
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