It is often desirable or even necessary to use more than one variable to model a situation in a field such as business, science, psychology, engineering, education, and sociology, to name a few. 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 is an equation equivalent to the equation
Joe Avalos has taken three exams in precalculus this semester. His average grade is 84. The highest grade is 18 points more than the average of the two lowers grades, and his lowest grade is 15 points less than the average of the two higher grades. What did he make on each exam?
There are three unknowns:
The first sentence can be rewritten as
It is going to get boring if we keep repeating the phrases
The first sentence
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 L in equation (1).
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 M first..
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 L, the second column contains the coefficients of M, the third column contains the coefficients of H, 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 Row 1 to Row 2 to form a new Row 2, and add 2 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