Key Terms

Key Terms

addition method

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

augmented matrix

a coefficient matrix adjoined with the constant column separated by a vertical line within the matrix brackets

break-even point

the point at which a cost function intersects a revenue function; where profit is zero

coefficient matrix

a matrix that contains only the coefficients from a system of equations

column

a set of numbers aligned vertically in a matrix

consistent system

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

cost function

the function used to calculate the costs of doing business; it usually has two parts, fixed costs and variable costs

Cramer’s Rule

a method for solving systems of equations that have the same number of equations as variables using determinants

dependent system

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

determinant

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

entry

an element, coefficient, or constant in a matrix

feasible region

the solution to a system of nonlinear inequalities that is the region of the graph where the shaded regions of each inequality intersect

Gaussian elimination

using elementary row operations to obtain a matrix in row-echelon form

identity matrix

a square matrix containing ones down the main diagonal and zeros everywhere else; it acts as a 1 in matrix algebra

inconsistent system

a system of linear equations with no common solution because they represent parallel lines, which have no point or line in common

independent system

a system of linear equations with exactly one solution pair $\left(x,y\right)$

main diagonal

entries from the upper left corner diagonally to the lower right corner of a square matrix

matrix

a rectangular array of numbers

multiplicative inverse of a matrix

a matrix that, when multiplied by the original, equals the identity matrix

nonlinear inequality

an inequality containing a nonlinear expression

partial fraction decomposition

the process of returning a simplified rational expression to its original form, a sum or difference of simpler rational expressions

partial fractions

the individual fractions that make up the sum or difference of a rational expression before combining them into a simplified rational expression

profit function

the profit function is written as $P(x)=R(x)-C(x),$ revenue minus cost

revenue function

the function that is used to calculate revenue, simply written as $R=xp,$ where $x=$ quantity and $p=$ price

row

a set of numbers aligned horizontally in a matrix

row operations

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

row-echelon form

after performing row operations, the matrix form that contains ones down the main diagonal and zeros at every space below the diagonal

row-equivalent

two matrices $A$ and $B$ are row-equivalent if one can be obtained from the other by performing basic row operations

scalar multiple

an entry of a matrix that has been multiplied by a scalar

solution set

the set of all ordered pairs or triples that satisfy all equations in a system of equations

substitution method

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

system of linear equations

a set of two or more equations in two or more variables that must be considered simultaneously.

system of nonlinear equations

a system of equations containing at least one equation that is of degree larger than one

system of nonlinear inequalities

a system of two or more inequalities in two or more variables containing at least one inequality that is not linear