It is often desirable or even necessary to use more than one variable to model situations in many fields. When this is the case, we write and solve a system of equations in order to answer questions about the situation.
If a system of linear equations has at least one solution, it is consistent. If the system has no solutions, it is inconsistent. If the system has an infinity number of solutions, it is dependent. Otherwise it is independent.
A linear equation in three variables describes a plane and is an equation equivalent to the equation
The ABC Shipping Company charges $2.90 for all packages weighing less than or equal to 5 lbs, $5.20 for packages weighting more than 5 lbs and less than 10 lbs, and $8.00 for all packages weighing 10 lbs or more. The number of packages weighing 5 lbs or less is 50% more than the number of packages weighing 10 lbs or more. One day shipping charges for 300 orders was $1,508. Find the number of packages in each category. .
There are three unknowns:
[ The number of packages weighing 5 lbs or less ] [ The number of packages weighing between 5 lbs and 10 lbs ] [ The number of packages weighing 10 lbs or more ] The $2.90 per package charge for [ The number of packages weighing 5 lbs or less ] the $5.20 per package charge for [ The number of packages weighing between 5 lbs and 10 lbs ] the $8.00 per package charge for [ The number of packages weighing 10 lbs or more ] [ The number of packages weighing 5 lbs or less ] times [ The number of packages weighing 10 lbs or more ]
It is going to get boring if we keep repeating the phrases
The sentence [ The number of packages weighing 5 lbs or less ] [ The number of packages weighing between 5 lbs and 10 lbs ] [ The number of packages weighing 10 lbs or more ] can now be written as
We have converted the problem from one described by words to one that is described by three equations.
1) Substitution, 2) Elimination 3) Matrices
The process of substitution involves several steps:
Step 1: Solve for one of the variables in one of the equations. It makes no difference which equation and which variable you choose. Let's solve for in equation (3) because the equation only has two variables.
The process of elimination involves several steps: First you reduce three equations to two equations with two variables, and then to one equation with one variable.
Step 1: Decide which variable you will eliminate. It makes no difference which one you choose. Let us eliminate first because is missing from equation (3).
The process of using matrices is essentially a shortcut of the process of elimination. Each row of the matrix represents an equation and each column represents coefficients of one of the variables.
Step 1: Create a three-row by four-column matrix using coefficients and the constant of each equation.
The vertical lines in the matrix stands for the equal signs between both sides of each equation. The first column contains the coefficients of x, the second column contains the coefficients of y, the third column contains the coefficients of z, and the last column contains the constants.
We want to convert the original matrix
Step 2: We work with column 1 first. The number 1 is already in cell 11(Row1-Col 1). Add times Row 1 to Row 2 to form a new Row 2, and add times Row 1 to Row 3 to form a new Row 3..
If you would like to work a similar example, click on Example.
If you would like to test yourself by working some problem similar to this example, click on Problem.
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Author: Nancy Marcus