Chapter 2 – Systems of Equations and Inequalities

2.1 Solving Systems of Equations Definitions A consistent system of equations has at least one solution. If there is exactly one solution, the system is independent. If there are infinitely many solutions the system is dependent. If the lines are parallel, don’t intersect, then the system is said to be inconsistent.

Solve the following system by graphing: Solve the following system using elimination:

Solve the following system using substitution:

HomeMade Toys manufacturers solid pine trucks and cars and usually sells four times as many trucks as cars. The net profit from each truck is $6 and from each car, $5. If the company wants a total profit of $29,000, how many trucks and cars should they sell? The Music Booster Club of Central High sells CD’s and tapes to raise money. During the last fund raiser 48 CD’s and 32 tapes sold for $1,040. This time they hope to sell three times as many CD’s and four times as many tapes for $3,440. What is the selling price of each?

2.2 Introducing Matrices The following display is called a matrix. A matrix is any rectangular array of terms called elements. The elements of a matrix are arranged in rows and columns. A matrix with m rows and n columns is an m x n matrix. Those are also the dimensions, m and n. Row matrix Column matrix Square matrix

Def: Two matrices are equal iff they have the same dimensions and are identical, element by element. Example: Find the values of x and y for which the following equation is true. How to add matrices:Zero Matrix: Example: Find A + B if and

EX: Find A – B if and A matrix can be multiplied my a scalar. A scalar is a real number that can multiply with a matrix. Example: Rules for matrix multiplication:

Ex: Find AB if and

2.3 Determinates and Multiplicative Inverses of Matrices Each square matrix has a determinant, a specific type of function. The value of is This works for 2x2 matrices. Ex: Find

Discussion on a minor and how to calculate one. Third-Order Determinant:

Example: Find the value of two ways. What is the identity matrix?

If and, then Inverse of a Second-Order Matrix

Example: Find the multiplicative inverse of Example: Solve the following system by using matrix multiplication

2.4 Solving Systems of Equations by Using Matrices What is an augmented matrix? And how are we going to use it to solve equations? Row Operations on Matrices: 1.Interchange any two rows 2.Replace any row with a nonzero multiple of that row 3.Replace any row with the sum of the row and a multiple of another row

Solve the system of equations by using row operations:

Example: Find the equation of a parabola that contains the points at (1,9), (4,6) and (6,14).

2.5 Solving Systems of Linear Inequalities Solve the following systems of inequalities by graphing:

Solve the linear inequality by graphing: What is linear programming and how does it maximize or minimize a problem?