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|Finite mathematics topic summary: systems of linear equations and matrices
Linear Equations in Two Unknowns
A linear equation in two unknowns is an equation that can be written in the form
with a, b and c being real numbers, and with a and b not both zero.
The graph of such an equation is a straight line. (See the Topic 1 Summary for a summary of straight lines.)
The following equations are linear:
the following equations are not linear:
Solutions of Two Linear Equations in Two Unknowns
A solution to a linear equation in the two unknowns x and y consists of a pair of numbers: a value for x and a value for y that satisfy the equation. By the same token, a solution to a system of two or more linear equations in x and y is a solution that satisfies all of the equations in the system.
We can solve such a system of equations either graphically, by drawing the graphs and finding where they intersect, or algebraically, by combining the equations in order to eliminate all but one variable and then solving for that variable.
A system of two linear equations in two unknowns has either:
(1) A single (or unique) solution. This happens when the lines corresponding to the two equations are not parallel, so that they intersect at a single point.
(2) No solution. This happens when the two lines are parallel and different.
(3) An infinite number of solutions. This occurs when the two equations represent the same straight line. In this case, we represent the solutions by choosing one variable arbitrarily, and solving for the other.
In the linear equation
The augmented matrix of a system of linear equations is the matrix whose rows are the coefficients of the equations together with the right-hand sides.
Go to the tutorial for this topic for some practice in setting up the augmented matrix.
Gauss Jordan Row Reduction
Following are the elementary row operations we use:
1. Replacing Ri by aRi where a is a nonzero number (in words: multiplying or dividing a row by a non-zero number).
Using operations of these three types, it is possible to row reduce any matrix. A matrix is row-reduced, or in reduced row echelon form if:
P1. The first non zero entry in each row (called the leading entry of that row) is a 1.
The procedure of row reduction is also known as Gauss-Jordan reduction.
Go to the tutorial for some practice in doing row operations.
Multiplying a row by a non-zero number
Replacing a row by a combination with another row
Switching two rows
Consistent and Inconsistent Systems
A system of linear equations either has no solutions, a unique solution, or an infinite number of solutions. If it has solutions it is said to be consistent, otherwise it is inconsistent. A system of linear equations in which there are fewer equations than unknowns is said to be underdetermined. These are the systems that often give infinitely many solutions. A system of equations in which the number of equations exceeds the number of unknowns is said to be overdetermined. In an overdetermined system, anything can happen, but such a system will often be inconsistent.