Comprehensive Algebra, Matrices, and Probability Review for Math 121 Questions fully solved & verified for accuracy(A+graded)

What are systems of equations?

A set of equations with the same variables that are solved simultaneously.

What is graphing in relation to systems of equations?

A method of solving systems of equations by plotting the equations on a graph and finding their intersection point.

What is substitution in solving systems of equations?

A method where one equation is solved for one variable, and that expression is substituted into another equation.

What is elimination in solving systems of equations?

A method that involves adding or subtracting equations to eliminate one variable, making it easier to solve for the other variable.

What does it mean if a system of equations has no solution?

It means the equations represent parallel lines that never intersect.

What does it mean if a system of equations has an infinite number of solutions?

It means the equations represent the same line, or overlapping lines.

What is an augmented matrix?

A matrix that includes the coefficients and constants of a system of equations.

What are the three row operations in matrix manipulation?

1. Swapping two rows, 2. Multiplying a row by a non-zero scalar, 3. Adding or subtracting rows.

What is Gauss-Jordan elimination?

A method for solving systems of equations by transforming the augmented matrix into reduced row echelon form.

What is an identity matrix?

A square matrix with ones on the diagonal and zeros elsewhere.

What is the Blueprint Method?

A systematic approach for solving systems of equations using Gauss-Jordan elimination.

What is a feasible region in linear inequalities?

The set of all possible solutions that satisfy the constraints of a linear inequality.

What is linear programming?

A method for finding the maximum or minimum value of a linear function subject to constraints.

What is a corner point in linear programming?

A point where the constraints of the feasible region intersect, which may represent optimal solutions.

What is a set in mathematics?

A collection of distinct objects, considered as an object in its own right.

What is a subset?

A set where every element is also contained in another set.

What is the union of two sets?

The set containing all elements from both sets, without duplicates.

What is the intersection of two sets?

The set containing only the elements that are in both sets.

What is a factorial?

The product of all positive integers up to a given number, denoted as n!.

What is a probability experiment?

An action or process that leads to one or more outcomes.

What is a sample space?

The set of all possible outcomes of a probability experiment.

What is conditional probability?

The probability of an event occurring given that another event has already occurred.

What is Bayes' Equation?

A formula that describes the probability of an event based on prior knowledge of conditions related to the event.

What is a random variable?

A variable whose possible values are numerical outcomes of a random phenomenon.

What is a probability distribution?

A function that describes the likelihood of obtaining the possible values that a random variable can take.

What is a binomial experiment?

A probability experiment that has a fixed number of trials, each with two possible outcomes.

What is a Markov Chain?

A stochastic process that undergoes transitions from one state to another on a state space.