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Key Terms
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addition method
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an algebraic technique used to solve systems of linear equations in which the equations are added in a way that eliminates one variable, allowing the resulting equation to be solved for the remaining variable; substitution is then used to solve for the first variable
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augmented matrix
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a coefficient matrix adjoined with the constant column separated by a vertical line within the matrix brackets
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break-even point
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the point at which a cost function intersects a revenue function; where profit is zero
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coefficient matrix
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a matrix that contains only the coefficients from a system of equations
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column
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a set of numbers aligned vertically in a matrix
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consistent system
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a system for which there is a single solution to all equations in the system and it is an independent system, or if there are an infinite number of solutions and it is a dependent system
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cost function
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the function used to calculate the costs of doing business; it usually has two parts, fixed costs and variable costs
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Cramer’s Rule
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a method for solving systems of equations that have the same number of equations as variables using determinants
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dependent system
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a system of linear equations in which the two equations represent the same line; there are an infinite number of solutions to a dependent system
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determinant
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a number calculated using the entries of a square matrix that determines such information as whether there is a solution to a system of equations
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entry
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an element, coefficient, or constant in a matrix
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feasible region
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the solution to a system of nonlinear inequalities that is the region of the graph where the shaded regions of each inequality intersect
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Gaussian elimination
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using elementary row operations to obtain a matrix in row-echelon form
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identity matrix
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a square matrix containing ones down the main diagonal and zeros everywhere else; it acts as a 1 in matrix algebra
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inconsistent system
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a system of linear equations with no common solution because they represent parallel lines, which have no point or line in common
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independent system
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a system of linear equations with exactly one solution pair
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main diagonal
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entries from the upper left corner diagonally to the lower right corner of a square matrix
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matrix
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a rectangular array of numbers
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multiplicative inverse of a matrix
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a matrix that, when multiplied by the original, equals the identity matrix
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nonlinear inequality
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an inequality containing a nonlinear expression
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partial fraction decomposition
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the process of returning a simplified rational expression to its original form, a sum or difference of simpler rational expressions
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partial fractions
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the individual fractions that make up the sum or difference of a rational expression before combining them into a simplified rational expression
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profit function
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the profit function is written as
revenue minus cost
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revenue function
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the function that is used to calculate revenue, simply written as
where
quantity and
price
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row
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a set of numbers aligned horizontally in a matrix
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row operations
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adding one row to another row, multiplying a row by a constant, interchanging rows, and so on, with the goal of achieving row-echelon form
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row-echelon form
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after performing row operations, the matrix form that contains ones down the main diagonal and zeros at every space below the diagonal
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row-equivalent
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two matrices
and
are row-equivalent if one can be obtained from the other by performing basic row operations
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scalar multiple
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an entry of a matrix that has been multiplied by a scalar
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solution set
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the set of all ordered pairs or triples that satisfy all equations in a system of equations
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substitution method
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an algebraic technique used to solve systems of linear equations in which one of the two equations is solved for one variable and then substituted into the second equation to solve for the second variable
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system of linear equations
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a set of two or more equations in two or more variables that must be considered simultaneously.
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system of nonlinear equations
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a system of equations containing at least one equation that is of degree larger than one
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system of nonlinear inequalities
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a system of two or more inequalities in two or more variables containing at least one inequality that is not linear