Mr. Hartzer, Hamtramck High School

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Mr. Hartzer, Hamtramck High School 2017-18 Algebra II Mr. Hartzer, Hamtramck High School 2017-18

DO NOW Continue working on the review packet.

This Week’s Objective Ch. 3 Students will: review solving linear systems and basic matrix operations.

Today’s Objective Ch. 3 I will be able to: review solving linear systems and basic matrix operations.

Today’s Agenda Review slides Practice exercises

System of Equations A system of equations is a set of equations that have the same variables. A solution is an ordered set of values that satisfy all the equations.

System of Equations A linear system is a system of two linear equations. The solution to a linear system is the point (ordered pair) of intersection between the two lines.

System of Equations To solve a system of two equations graphically, graph each equation and look for any points of intersections.

Solving by Substitution To solve by substitution: (1) Solve one equation for one of the variables. (2) Substitute the variable for the expression in the other equation. (3) Solve for the value of the variable. (4) Use this value in the first equation to solve for the value of the other variable.

Solving by Elimination To solve by elimination: (1) Multiply one or both equations to create matching coefficients. (2) Take the difference of the equations to eliminate one variable. (3) Solve for the value of the other variable. (4) Use this value in either equation to solve for the value of the first variable.

Systems of Linear Inequalities The solutions of a system of linear inequalities are all points that satisfy all inequalities. Usually, this is a shaded area, not a single point.

Systems of Linear Equations A system of three linear equations has three variables and three equations. A solution is an ordered triple (x, y, z) whose coordinates make each equation true.

Systems of Linear Equations To solve a system of three variables: Pair up equations to eliminate one variable. Solve the new two-variable equations. Use the two values to solve for the third. ORGANIZATION IS CRUCIAL!

Matrix Basics A matrix is a rectangular array of numeric expressions.

Matrix Basics Example matrix A is a 2 x 3 matrix. Always list the height FIRST. Use brackets to indicate a matrix.

Matrix Basics Each expression is called an element. 𝑎 12 =4: The Element in row 1, col 2 is 4.

Matrix Operations Addition/subtraction: To add matrices A and B with the same dimensions, add corresponding elements. Scalar multiplication: To multiple a constant c with A, multiply c with each element of A.

Matrix Multiplication KEY FACT! We can only multiply two matrices, A and B, if the number of A’s columns is the same as the number of B’s rows.

Matrix Multiplication KEY CONCEPT! Multiply the ROWS of the LEFT MATRIX with the COLUMNS of the RIGHT MATRIX.

Matrix Multiplication: 2x2 Model 1 2 3 4 5 7 6 8 = 1⋅5+2⋅6 1⋅7+2⋅8 3⋅5+4⋅6 3⋅7+4⋅8 = 5+12 7+16 15+24 21+32 = 17 23 39 53

Augmented Matrix An augmented matrix contains the coefficients of a linear system: 3𝑥+𝑦=7 −𝑥=5 3 1 −1 0 7 5

Row Operations Goal: Use row operations to create 1 0 𝑥 0 1 𝑦 Multiply (or divide) a single row by a constant. Add (or subtract) one row from another.