Put the following equations to slope-intercept form then use 2 points to graph
|
|
|
- Bertha Ball
- 5 years ago
- Views:
Transcription
1 Tuesday September 23, 2014 Warm-up: Put the following equations to slope-intercept form then use 2 points to graph 1. 4x - 3y = 8 8 x 6y = x + y = 4 2x + y = 1
2 Tuesday September 23, 2014 Warm-up: Put the following equations to slope-intercept form then use 2 points to graph 1. 4x - 3y = 8 8 x 6y = x + y = 4 2x + y = 1
3 Tuesday September 23, Solving Linear Systems using Graph Objective: To solve a system of Linear equations EQ: How many ways can we solve a system of linear equations Definition: A system of linear equations consists of 2 or more equations The solution of a system of linear equations can be 1 of the following 3 cases: 1. Exactly 1 solution Example1: When slope (m) and y-intercept (b) are given Lines intersect at 1 point Write the equation for the line in (consistent the graph and independent) Solution: Use Slope-Intercept form: According 2. Infinitely to the graph, many b solutions = -1 and m = ( ) / (3 0) = -2/3 Lines coincide Therefore, the equation is y = (-2/3) x - 1 (consistent and dependent) 3. No solutions Lines are parallel (inconsistent)
4 3.1 Solving Linear Systems using Graph There are 4 ways to solve a system of linear equations 1. Graphing 2. Substitution 3. Elimination 4. Multiplication/Elimination Steps to solve a linear system using graphing: 1. Put each equation in slope-intercept form (y = mx+b) 2. Use y-intercept to plot 1 st point and use m to plot 2 nd point 3. Look for solution(s) 4. Substitute the solution in each equation to check for error Example: Graph the linear system and estimate the solution then check the solution algebraically 4x + y = 8 (1) 2x 3y = 18 (2) Solution: 1. Put equations in y = mx + b form y = -4x + 8 (1) y = (2/3)x - 6 (2) 2. In (1), m = -4, b = 8 In (2), m = 2/3, b = 6 3. Solution is (3, -4) 4. Check: Tuesday September 23, 2014
5 Warm-up: Wednesday September 24, 2014
6 Example: Solve the system using the substitution method 2x + 5y = -5 (1) x + 3y = 3 (2) Solution: Step 1. Solve (2) for x x + 3y = 3 (2) x = -3y + 3 (3) Step 2. Substitute (3) into (1) 2x + 5y = -5 (1) 2 (-3y +3) + 5 y = -5-6 y y = y = -5 y = 11 Wednesday September 24, Solving Linear Systems by Substitution Steps to solve a linear system using substitution method: 1. Solve one of the equation for one of its variables 2. Substitute the expression from step 1 into the other equation and solve for the other variable 3. Substitute the value from step 2 into the revised equation from step 1 and solve for the other variable Step 3. Substitute y =11 back to either (1) or (2) x + 3y = 3 (2) x + 3 (11) = 3 x + 33 = 3 x = -30 The solution is (-30, 11) Check:
7 Example: Solve the system using the elimination method 6x - 14y = 20 (1) 6x - 8y = 8 (2) Solution: Step 1. Subtract (1) and (2) to eliminate x 6x - 14y = 20 (1) 6x - 8y = 8 (2) y = 12 y = - 2 Wednesday September 24, Solving Linear Systems by Elimination Steps to solve a linear system using elimination method: 1. Add or subtract the equations to eliminate one of the variables. Then solve for the other variable 2. Substitute the expression from step 1 into either the original equation and solve for the other variable. Step 2. Substitute y = -2 into either (1) or (2) 6x - 8y = 8 (2) 6x - 8 (-2) = 8 6x + 16 = 8 6x = x = -8 x = -8/6 = -4/3 The solution is (-4/3, -2) Check: (1) 6(-4/3) - 14 (-2) = 20 (2) 6(-4/3) 8 (-2) = = = 8 Yes Yes
8 Wednesday September 24, Solving Linear Systems by Multiplication and Elimination Steps to solve a linear system using multiplication and elimination method: 1. Multiply on or both of the equation by a constant to make the coefficient of one variable in both equations the same 2. Add or subtract the equations to eliminate one of the variables. Then solve for the other variable 3. Substitute the expression from step 1 into either the original equation and solve for the other variable. Example: Solve the system using the elimination method 3x - 7y = 10 (1) 6x - 8y = 8 (2) Solution: Step 1. Multiply (1) with 2 2 (3x - 7y = 10) x - 14y = 20 Step 2. Subtract (1) and (2) to eliminate x 6x - 14y = 20 (1) 6x - 8y = 8 (2) y = 12 y = - 2 Step 2. Substitute y = -2 into either (1) or (2) 6x - 8y = 8 (2) 6x - 8 (-2) = 8 6x + 16 = 8 6x = x = -8 x = -8/6 = -4/3 The solution is (-4/3, -2) Check: (1) 6(-4/3) - 14 (-2) = 20 (2) 6(-4/3) 8 (-2) = = = 8 Yes Yes
9 3.5 Matrices-Basic Operations: Matrix is a rectangular arrangement of numbers in row and columns. Dimension: m x n (row x column) Each number is an element Two matrices are equal iff: 1. Their dimensions are the same 2. Corresponding elements are equal Friday October 18, 2013 Objective: To perform basic operations with matrices EQ: How can we organize sports data? Adding and Subtracting Matrices (only when dimensions are the same) Simply add or subtract corresponding elements in the corresponding positions Example: 1. 2.
10 Friday October 18, Matrices-Basic Operations: Scalar Multiplication Simply multiply each element with the scalar Example:
11 Warm-up: Find x and y Monday October 20, 2014
12 Monday October 20, 2014 Warm-up: Find x and y Solution: Therefore: The solution is x = -2 and y = 4
13 3.6 Multiply Matrices: Monday October 20, 2013 Objective: To perform basic operations with matrices EQ: How can we organize sports data? The product of 2 matrices A and B is defined iff: the number of columns in A is equal to the number of rows in B Example: State whether the product AB is defined: 1. A: 5x2 and B:2x2 defined, AB: 5x2 3. A: 4x3 and B:3x2 defined, AB: 4x2 2. A: 3x2 and B:3x2 undefined 4. A: 3x4 and B:3x2 undefined
14 3.6 Multiply Matrices: Monday October 21, 2013 Multiplying Matrices Multiply each element in the ith row of A to jth column of B Note: The product AB BA Example: Find product AB then find product BA if Solution: Find AB: since A: 2x2 and B: 2x2, AB: 2x2 1. AB = 2. AB = 3. AB = 4. AB = 5. AB = Find BA:
15 3.6 Multiply Matrices: Monday October 20, 2013 Example: Use matrices to calculate the total cost of 2 hockey teams. Sticks ($60) Pucks ($2) Women steam Men s Team Uniforms ($35) Solution: Let s write the equipment list and the cost list in matrix form: So, the total cost of equipment for each team can be found by multiply the equipment matrix (E) by the cost matrix (C ). Since E: 2x3 and C: 3x1, then EC: 2x1 The total cost for women s team is $1530 The total cost for men s team is $1710
Algebra 1 Semester 2 Final Review
Team Awesome 011 Name: Date: Period: Algebra 1 Semester Final Review 1. Given y mx b what does m represent? What does b represent?. What axis is generally used for x?. What axis is generally used for y?
Review for Mastery Using Graphs and Tables to Solve Linear Systems
3-1 Using Graphs and Tables to Solve Linear Systems A linear system of equations is a set of two or more linear equations. To solve a linear system, find all the ordered pairs (x, y) that make both equations
3-6 Lines in the Coordinate Plane
3-6 Lines in the Coordinate Plane Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up Substitute the given values of m, x, and y into the equation y = mx + b and solve for b. 1. m = 2, x = 3, and
10/26/ Solving Systems of Linear Equations Using Matrices. Objectives. Matrices
6.1 Solving Systems of Linear Equations Using Matrices Objectives Write the augmented matrix for a linear system. Perform matrix row operations. Use matrices and Gaussian elimination to solve systems.
Section 1.5. Finding Linear Equations
Section 1.5 Finding Linear Equations Using Slope and a Point to Find an Equation of a Line Example Find an equation of a line that has slope m = 3 and contains the point (2, 5). Solution Substitute m =
For example, the system. 22 may be represented by the augmented matrix
Matrix Solutions to Linear Systems A matrix is a rectangular array of elements. o An array is a systematic arrangement of numbers or symbols in rows and columns. Matrices (the plural of matrix) may be
HFCC Math Lab Intermediate Algebra 1 SLOPE INTERCEPT AND POINT-SLOPE FORMS OF THE LINE
HFCC Math Lab Intermediate Algebra SLOPE INTERCEPT AND POINT-SLOPE FORMS OF THE LINE THE EQUATION OF A LINE Goal I. Use the slope-intercept form of the line to write the equation of a non-vertical line
Instructor: Barry McQuarrie Page 1 of 6
Questions 1. Solve the system by graphing: 3x + y = 2 2x y = 3 2. Solve the system by graphing: x + 3y = 9 y = 1 3 x 2 3. Solve the system by graphing: y = 2x + 5 3y + 6x = 15 4. Solve the system algebraically,
Section 2.0: Getting Started
Solving Linear Equations: Graphically Tabular/Numerical Solution Algebraically Section 2.0: Getting Started Example #1 on page 128. Solve the equation 3x 9 = 3 graphically. Intersection X=4 Y=3 We are
Use the graph shown to determine whether each system is consistent or inconsistent and if it is independent or dependent.
Use the graph shown to determine whether each system is consistent or inconsistent and if it is independent or dependent. 12. y = 3x + 4 y = 3x 4 These two equations do not intersect, so they are inconsistent.
Vertical Line Test a relationship is a function, if NO vertical line intersects the graph more than once
Algebra 2 Chapter 2 Domain input values, X (x, y) Range output values, Y (x, y) Function For each input, there is exactly one output Example: Vertical Line Test a relationship is a function, if NO vertical
Name Hr. Honors Geometry Lesson 9-1: Translate Figures and Use Vectors
Name Hr Honors Geometry Lesson 9-1: Translate Figures and Use Vectors Learning Target: By the end of today s lesson we will be able to successfully use a vector to translate a figure. Isometry: An isometry
Matrices. A Matrix (This one has 2 Rows and 3 Columns) To add two matrices: add the numbers in the matching positions:
Matrices A Matrix is an array of numbers: We talk about one matrix, or several matrices. There are many things we can do with them... Adding A Matrix (This one has 2 Rows and 3 Columns) To add two matrices:
Today is the last day to register for CU Succeed account AND claim your account. Tuesday is the last day to register for my class
Today is the last day to register for CU Succeed account AND claim your account. Tuesday is the last day to register for my class Back board says your name if you are on my roster. I need parent financial
3. Replace any row by the sum of that row and a constant multiple of any other row.
Math Section. Section.: Solving Systems of Linear Equations Using Matrices As you may recall from College Algebra or Section., you can solve a system of linear equations in two variables easily by applying
1. Answer: x or x. Explanation Set up the two equations, then solve each equation. x. Check
Thinkwell s Placement Test 5 Answer Key If you answered 7 or more Test 5 questions correctly, we recommend Thinkwell's Algebra. If you answered fewer than 7 Test 5 questions correctly, we recommend Thinkwell's
x = 12 x = 12 1x = 16
2.2 - The Inverse of a Matrix We've seen how to add matrices, multiply them by scalars, subtract them, and multiply one matrix by another. The question naturally arises: Can we divide one matrix by another?
Practice Test (page 391) 1. For each line, count squares on the grid to determine the rise and the run. Use slope = rise
Practice Test (page 91) 1. For each line, count squares on the grid to determine the rise and the. Use slope = rise 4 Slope of AB =, or 6 Slope of CD = 6 9, or Slope of EF = 6, or 4 Slope of GH = 6 4,
slope rise run Definition of Slope
The Slope of a Line Mathematicians have developed a useful measure of the steepness of a line, called the slope of the line. Slope compares the vertical change (the rise) to the horizontal change (the
Section 7D Systems of Linear Equations
Section 7D Systems of Linear Equations Companies often look at more than one equation of a line when analyzing how their business is doing. For example a company might look at a cost equation and a profit
REVIEW FOR THE FIRST SEMESTER EXAM
Algebra II Honors @ Name Period Date REVIEW FOR THE FIRST SEMESTER EXAM You must NEATLY show ALL of your work ON SEPARATE PAPER in order to receive full credit! All graphs must be done on GRAPH PAPER!
Systems of Inequalities and Linear Programming 5.7 Properties of Matrices 5.8 Matrix Inverses
5 5 Systems and Matrices Systems and Matrices 5.6 Systems of Inequalities and Linear Programming 5.7 Properties of Matrices 5.8 Matrix Inverses Sections 5.6 5.8 2008 Pearson Addison-Wesley. All rights
3.1 INTRODUCTION TO THE FAMILY OF QUADRATIC FUNCTIONS
3.1 INTRODUCTION TO THE FAMILY OF QUADRATIC FUNCTIONS Finding the Zeros of a Quadratic Function Examples 1 and and more Find the zeros of f(x) = x x 6. Solution by Factoring f(x) = x x 6 = (x 3)(x + )
Therefore, after becoming familiar with the Matrix Method, you will be able to solve a system of two linear equations in four different ways.
Grade 9 IGCSE A1: Chapter 9 Matrices and Transformations Materials Needed: Straightedge, Graph Paper Exercise 1: Matrix Operations Matrices are used in Linear Algebra to solve systems of linear equations.
Finite Math - J-term Homework. Section Inverse of a Square Matrix
Section.5-77, 78, 79, 80 Finite Math - J-term 017 Lecture Notes - 1/19/017 Homework Section.6-9, 1, 1, 15, 17, 18, 1, 6, 9, 3, 37, 39, 1,, 5, 6, 55 Section 5.1-9, 11, 1, 13, 1, 17, 9, 30 Section.5 - Inverse
Intro. To Graphing Linear Equations
Intro. To Graphing Linear Equations The Coordinate Plane A. The coordinate plane has 4 quadrants. B. Each point in the coordinate plain has an x-coordinate (the abscissa) and a y-coordinate (the ordinate).
Vector: A series of scalars contained in a column or row. Dimensions: How many rows and columns a vector or matrix has.
ASSIGNMENT 0 Introduction to Linear Algebra (Basics of vectors and matrices) Due 3:30 PM, Tuesday, October 10 th. Assignments should be submitted via e-mail to: matlabfun.ucsd@gmail.com You can also submit
Precalculus Notes: Unit 7 Systems of Equations and Matrices
Date: 7.1, 7. Solving Systems of Equations: Graphing, Substitution, Elimination Syllabus Objectives: 8.1 The student will solve a given system of equations or system of inequalities. Solution of a System
College Prep Algebra II Summer Packet
Name: College Prep Algebra II Summer Packet This packet is an optional review which is highly recommended before entering CP Algebra II. It provides practice for necessary Algebra I topics. Remember: When
Course Number 432/433 Title Algebra II (A & B) H Grade # of Days 120
Whitman-Hanson Regional High School provides all students with a high- quality education in order to develop reflective, concerned citizens and contributing members of the global community. Course Number
2.1. Rectangular Coordinates and Graphs. 2.1 Rectangular Coordinates and Graphs 2.2 Circles 2.3 Functions 2.4 Linear Functions. Graphs and Functions
2 Graphs and Functions 2 Graphs and Functions 2.1 Rectangular Coordinates and Graphs 2.2 Circles 2.3 Functions 2.4 Linear Functions Sections 2.1 2.4 2008 Pearson Addison-Wesley. All rights reserved Copyright
Chapter 1 Section 1 Solving Linear Equations in One Variable
Chapter Section Solving Linear Equations in One Variable A linear equation in one variable is an equation which can be written in the form: ax + b = c for a, b, and c real numbers with a 0. Linear equations
SYSTEMS OF LINEAR EQUATIONS
SYSTEMS OF LINEAR EQUATIONS A system of linear equations is a set of two equations of lines. A solution of a system of linear equations is the set of ordered pairs that makes each equation true. That is
GEOMETRY HONORS COORDINATE GEOMETRY PACKET
GEOMETRY HONORS COORDINATE GEOMETRY PACKET Name Period Homework Lesson Assignment Day 1 - Slopes of Perpendicular WKSHT and Parallel Lines Day 2 - Writing an Equation of a Line HW- Honors TXTBK pages 615-617
Maths for Signals and Systems Linear Algebra in Engineering. Some problems by Gilbert Strang
Maths for Signals and Systems Linear Algebra in Engineering Some problems by Gilbert Strang Problems. Consider u, v, w to be non-zero vectors in R 7. These vectors span a vector space. What are the possible
Solving Systems of Equations Using Matrices With the TI-83 or TI-84
Solving Systems of Equations Using Matrices With the TI-83 or TI-84 Dimensions of a matrix: The dimensions of a matrix are the number of rows by the number of columns in the matrix. rows x columns *rows
Chapter 1. Linear Equations and Straight Lines. 2 of 71. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1 Linear Equations and Straight Lines 2 of 71 Outline 1.1 Coordinate Systems and Graphs 1.4 The Slope of a Straight Line 1.3 The Intersection Point of a Pair of Lines 1.2 Linear Inequalities 1.5
Graphing Linear Equations
Graphing Linear Equations Question 1: What is a rectangular coordinate system? Answer 1: The rectangular coordinate system is used to graph points and equations. To create the rectangular coordinate system,
ALGEBRA 2 W/ TRIGONOMETRY MIDTERM REVIEW
Name: Block: ALGEBRA W/ TRIGONOMETRY MIDTERM REVIEW Algebra 1 Review Find Slope and Rate of Change Graph Equations of Lines Write Equations of Lines Absolute Value Functions Transformations Piecewise Functions
GCSE-AS Mathematics Bridging Course. Chellaston School. Dr P. Leary (KS5 Coordinator) Monday Objectives. The Equation of a Line.
GCSE-AS Mathematics Bridging Course Chellaston School Dr (KS5 Coordinator) Monday Objectives The Equation of a Line Surds Linear Simultaneous Equations Tuesday Objectives Factorising Quadratics & Equations
3.5 Day 1 Warm Up. Graph each line. 3.4 Proofs with Perpendicular Lines
3.5 Day 1 Warm Up Graph each line. 1. y = 4x 2. y = 3x + 2 3. y = x 3 4. y = 4 x + 3 3 November 2, 2015 3.4 Proofs with Perpendicular Lines Geometry 3.5 Equations of Parallel and Perpendicular Lines Day
Matrix Inverse 2 ( 2) 1 = 2 1 2
Name: Matrix Inverse For Scalars, we have what is called a multiplicative identity. This means that if we have a scalar number, call it r, then r multiplied by the multiplicative identity equals r. Without
A Poorly Conditioned System. Matrix Form
Possibilities for Linear Systems of Equations A Poorly Conditioned System A Poorly Conditioned System Results No solution (inconsistent) Unique solution (consistent) Infinite number of solutions (consistent)
Geometry Unit 2: Linear. Section Page and Problems Date Assigned
Geometry Name: Geometry Unit 2: Linear Topics Covered: Midpoint formula Distance formula Slope Slope- Intercept Form Point- Slope Form Standard Form Assignment # Section Page and Problems Date Assigned
Graphs and Linear Functions
Graphs and Linear Functions A -dimensional graph is a visual representation of a relationship between two variables given by an equation or an inequality. Graphs help us solve algebraic problems by analysing
9.1: GRAPHING QUADRATICS ALGEBRA 1
9.1: GRAPHING QUADRATICS ALGEBRA 1 OBJECTIVES I will be able to graph quadratics: Given in Standard Form Given in Vertex Form Given in Intercept Form What does the graph of a quadratic look like? https://www.desmos.com/calculator
Section 18-1: Graphical Representation of Linear Equations and Functions
Section 18-1: Graphical Representation of Linear Equations and Functions Prepare a table of solutions and locate the solutions on a coordinate system: f(x) = 2x 5 Learning Outcome 2 Write x + 3 = 5 as
SIMULTANEOUS EQUATIONS
Mathematics Revision Guides Simultaneous Equations Page 1 of 6 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Foundation Tier SIMULTNEOUS EQUTIONS Version: 3.2 Date: 08-02-2015 Mathematics Revision
Lakeview Christian Academy Summer Math Packet For Students Entering Algebra 2
Lakeview Christian Academy Summer Math Packet For Students Entering Algebra Student s Name This packet is designed for you to review your Algebra 1 skills and make sure you are well prepared for the start
PRE-ALGEBRA BY MYRL SHIREMAN
PRE-ALGEBRA BY MYRL SHIREMAN COPYRIGHT 1994 Mark Twain Media, Inc. ISBN 10-digit: 1-58037-064-0 13-digit: 978-1-58037-064-6 Printing No. CD-1876 Mark Twain Media, Inc., Publishers Distributed by Carson-Dellosa
Matrices and Determinants
pr8-78-88.i-hr /6/6 : PM Page 78 CHAPTER 8 Matrices and Determinants J ARON LANIER, WHO FIRST USED the term virtual reality, is chief scientist for the teleimmersion project, which explores the impact
SNAP Centre Workshop. Graphing Lines
SNAP Centre Workshop Graphing Lines 45 Graphing a Line Using Test Values A simple way to linear equation involves finding test values, plotting the points on a coordinate plane, and connecting the points.
You should be able to plot points on the coordinate axis. You should know that the the midpoint of the line segment joining (x, y 1 1
Name GRAPHICAL REPRESENTATION OF DATA: You should be able to plot points on the coordinate axis. You should know that the the midpoint of the line segment joining (x, y 1 1 ) and (x, y ) is x1 x y1 y,.
Precalculus Summer Packet
Precalculus Summer Packet Name: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This packet is to help you review various topics that are considered to be prerequisite knowledge
8.NS.1 8.NS.2. 8.EE.7.a 8.EE.4 8.EE.5 8.EE.6
Standard 8.NS.1 8.NS.2 8.EE.1 8.EE.2 8.EE.3 8.EE.4 8.EE.5 8.EE.6 8.EE.7 8.EE.7.a Jackson County Core Curriculum Collaborative (JC4) 8th Grade Math Learning Targets in Student Friendly Language I can identify
Geometry Pre AP Graphing Linear Equations
Geometry Pre AP Graphing Linear Equations Name Date Period Find the x- and y-intercepts and slope of each equation. 1. y = -x 2. x + 3y = 6 3. x = 2 4. y = 0 5. y = 2x - 9 6. 18x 42 y = 210 Graph each
February 01, Matrix Row Operations 2016 ink.notebook. 6.6 Matrix Row Operations. Page 49 Page Row operations
6.6 Matrix Row Operations 2016 ink.notebook Page 49 Page 50 6.6 Row operations (Solve Systems with Matrices) Lesson Objectives Page 51 Standards Lesson Notes Page 52 6.6 Matrix Row Operations Press the
Mathematics (www.tiwariacademy.com)
() Miscellaneous Exercise on Chapter 10 Question 1: Find the values of k for which the line is (a) Parallel to the x-axis, (b) Parallel to the y-axis, (c) Passing through the origin. Answer 1: The given
The shortest distance from point K to line is the length of a segment perpendicular to from point K. Draw a perpendicular segment from K to.
8. Find the distance between each pair of parallel lines with the given equations. Copy each figure. Construct the segment that represents the distance indicated. 12. K to The shortest distance from point
Revision Problems for Examination 2 in Algebra 1
Centre for Mathematical Sciences Mathematics, Faculty of Science Revision Problems for Examination in Algebra. Let l be the line that passes through the point (5, 4, 4) and is at right angles to the plane
Math 1313 Prerequisites/Test 1 Review
Math 1313 Prerequisites/Test 1 Review Test 1 (Prerequisite Test) is the only exam that can be done from ANYWHERE online. Two attempts. See Online Assignments in your CASA account. Note the deadline too.
Algebra IA. Unit 1 Connections to Algebra
A Unit 1 Connections to Algebra Time: 20 days Objectives: 1, 2, 8 and 9 Translate verbal into mathematical Write using exponents Use the order of operations to evaluate open sentences by performing arithmetic
Practice Test - Chapter 6
1. Write each system of equations in triangular form using Gaussian elimination. Then solve the system. Align the variables on the left side of the equal sign. Eliminate the x-term from the 2nd equation.
Linear Functions. College Algebra
Linear Functions College Algebra Linear Function A linear function is a function whose graph is a straight line. Linear functions can be written in the slope-intercept form of a line: f(x) = mx + b where
Name: Date: Study Guide: Systems of Equations and Inequalities
Name: Date: Study Guide: Systems of Equations and Inequalities Systems of Equations Linear systems consist of two or more linear equations in the same variables. A solution to the linear system of equations
Revision Topic 11: Straight Line Graphs
Revision Topic : Straight Line Graphs The simplest way to draw a straight line graph is to produce a table of values. Example: Draw the lines y = x and y = 6 x. Table of values for y = x x y - - - - =
Do you need a worksheet or a copy of the teacher notes? Go to
Name Period Day Date Assignment (Due the next class meeting) Wednesday Thursday Friday Monday Tuesday Wednesday Thursday Friday Monday Tuesday Wednesday Thursday Friday Monday Tuesday Wednesday Thursday
Section 1.2. Graphing Linear Equations
Graphing Linear Equations Definition of Solution, Satisfy, and Solution Set Definition of Solution, Satisfy, and Solution Set Consider the equation y = 2x 5. Let s find y when x = 3. y = 2x 5 Original
Algebra 2 Common Core Summer Skills Packet
Algebra 2 Common Core Summer Skills Packet Our Purpose: Completion of this packet over the summer before beginning Algebra 2 will be of great value to helping students successfully meet the academic challenges
10-2 Circles. Warm Up Lesson Presentation Lesson Quiz. Holt Algebra2 2
10-2 Circles Warm Up Lesson Presentation Lesson Quiz Holt Algebra2 2 Warm Up Find the slope of the line that connects each pair of points. 1. (5, 7) and ( 1, 6) 1 6 2. (3, 4) and ( 4, 3) 1 Warm Up Find
0_PreCNotes17 18.notebook May 16, Chapter 12
Chapter 12 Notes BASIC MATRIX OPERATIONS Matrix (plural: Matrices) an n x m array of elements element a ij Example 1 a 21 = a 13 = Multiply Matrix by a Scalar Distribute scalar to all elements Addition
Mid-Chapter Quiz: Lessons 1-1 through 1-4
Determine whether each relation represents y as a function of x. 1. 3x + 7y = 21 This equation represents y as a function of x, because for every x-value there is exactly one corresponding y-value. The
Mathematics Scope & Sequence Algebra I
Mathematics Scope & Sequence 2016-17 Algebra I Revised: June 20, 2016 First Grading Period (24 ) Readiness Standard(s) Solving Equations and Inequalities A.5A solve linear equations in one variable, including
Reteaching Transforming Linear Functions
Name Date Class Transforming Linear Functions INV 6 You have graphed linear functions on the coordinate plane. Now you will investigate transformations of the parent function for a linear function, f(x)
The Rectangular Coordinate System and Equations of Lines. College Algebra
The Rectangular Coordinate System and Equations of Lines College Algebra Cartesian Coordinate System A grid system based on a two-dimensional plane with perpendicular axes: horizontal axis is the x-axis
Sec 4.1 Coordinates and Scatter Plots. Coordinate Plane: Formed by two real number lines that intersect at a right angle.
Algebra I Chapter 4 Notes Name Sec 4.1 Coordinates and Scatter Plots Coordinate Plane: Formed by two real number lines that intersect at a right angle. X-axis: The horizontal axis Y-axis: The vertical
Math: Question 10
1 of 1 9/22/2016 7:55 PM Math: Question 10 A carpenter has $60 with which to buy supplies. The carpenter needs to buy both nails and screws. Nails cost $12.99 per box, and screws cost $14.99 per box. If
Topic 1.6: Lines and Planes
Math 275 Notes (Ultman) Topic 1.6: Lines and Planes Textbook Section: 12.5 From the Toolbox (what you need from previous classes): Plotting points, sketching vectors. Be able to find the component form
Solving Algebraic Equations
Lesson 4. Solving Algebraic Equations 3 3 3 3 3 8 8 4 Add 3 to both sides. Divide both sides by. 4 gives the solution of the equation 3. Check: Substitute 4 for x into the original equation. 3 4 3 When
Example 1: Give the coordinates of the points on the graph.
Ordered Pairs Often, to get an idea of the behavior of an equation, we will make a picture that represents the solutions to the equation. A graph gives us that picture. The rectangular coordinate plane,
CHAPTER HERE S WHERE YOU LL FIND THESE APPLICATIONS:
CHAPTER 8 You are being drawn deeper into cyberspace, spending more time online each week. With constantly improving high-resolution images, cyberspace is reshaping your life by nourishing shared enthusiasms.
Topic 4: Vectors Markscheme 4.6 Intersection of Lines and Planes Paper 2
Topic : Vectors Markscheme. Intersection of Lines and Planes Paper. Using an elimination method, x y + z x y z x y x + y 8 y Therefore x, y, z Using matrices, x y z x y z 5 (using a graphic display calculator)
The x-intercept can be found by setting y = 0 and solving for x: 16 3, 0
y=-3/4x+4 and y=2 x I need to graph the functions so I can clearly describe the graphs Specifically mention any key points on the graphs, including intercepts, vertex, or start/end points. What is the
January 24, Matrix Row Operations 2017 ink.notebook. 6.6 Matrix Row Operations. Page 35 Page Row operations
6.6 Matrix Row Operations 2017 ink.notebook Page 35 Page 36 6.6 Row operations (Solve Systems with Matrices) Lesson Objectives Page 37 Standards Lesson Notes Page 38 6.6 Matrix Row Operations Press the
Unit 2 Day 11. Quadratic Inequalities & Systems
Unit Day 11 Quadratic Inequalities & Systems 1 Warm Up 1. Given the following functions, specifically describe the transformation from the identity function y = x a. y = (x+3) 7 b. y = 5x + 1 c. y = ½(x
MATH 021 UNIT 2 HOMEWORK ASSIGNMENTS
MATH 021 UNIT 2 HOMEWORK ASSIGNMENTS General Instructions You will notice that most of the homework assignments for a section have more than one part. Usually, the part (A) questions ask for explanations,
To add or subtract, just add or subtract the numbers in the same column and row and place answer accordingly.
Math 3 Variable Manipulation Part 2 Systems with Matrices MATRICES An alternative method to solving system of equations is using Matrices. However, before we can solve systems of equations using matrices,
Solving Systems Using Row Operations 1 Name
The three usual methods of solving a system of equations are graphing, elimination, and substitution. While these methods are excellent, they can be difficult to use when dealing with three or more variables.
,!7IA3C1-cjfcei!:t;K;k;K;k ISBN Graphing Calculator Reference Card. Addison-Wesley s. Basics. Created in conjuction with
Addison-Wesley s Graphing Calculator Reference Card Created in conjuction with Basics Converting Fractions to Decimals The calculator will automatically convert a fraction to a decimal. Type in a fraction,
Chapter 1: Number and Operations
Chapter 1: Number and Operations 1.1 Order of operations When simplifying algebraic expressions we use the following order: 1. Perform operations within a parenthesis. 2. Evaluate exponents. 3. Multiply
5. In the Cartesian plane, a line runs through the points (5, 6) and (-2, -2). What is the slope of the line?
Slope review Using two points to find the slope In mathematics, the slope of a line is often called m. We can find the slope if we have two points on the line. We'll call the first point and the second
SLOPE A MEASURE OF STEEPNESS through 7.1.5
SLOPE A MEASURE OF STEEPNESS 7.1. through 7.1.5 Students have used the equation = m + b throughout this course to graph lines and describe patterns. When the equation is written in -form, the m is the
2-5 Rational Functions
Find the domain of each function and the equations of the vertical or horizontal asymptotes, if any. 3. f (x) = The function is undefined at the real zeros of the denominator b(x) = (x + 3)(x 4). The real
X Std. Topic Content Expected Learning Outcomes Mode of Transaction
X Std COMMON SYLLABUS 2009 - MATHEMATICS I. Theory of Sets ii. Properties of operations on sets iii. De Morgan s lawsverification using example Venn diagram iv. Formula for n( AÈBÈ C) v. Functions To revise
CW High School. Algebra I A
1. Functions (20.00%) 1.1 I can solve a two or more step equation involving parenthesis and negative numbers including those with no solution or all real numbers solutions. 4 Pro cient I can solve a two
Sketching graphs of polynomials
Sketching graphs of polynomials We want to draw the graphs of polynomial functions y = f(x). The degree of a polynomial in one variable x is the highest power of x that remains after terms have been collected.
MATH 423 Linear Algebra II Lecture 17: Reduced row echelon form (continued). Determinant of a matrix.
MATH 423 Linear Algebra II Lecture 17: Reduced row echelon form (continued). Determinant of a matrix. Row echelon form A matrix is said to be in the row echelon form if the leading entries shift to the
Four Types of Slope Positive Slope Negative Slope Zero Slope Undefined Slope Slope Dude will help us understand the 4 types of slope
Four Types of Slope Positive Slope Negative Slope Zero Slope Undefined Slope Slope Dude will help us understand the 4 types of slope https://www.youtube.com/watch?v=avs6c6_kvxm Direct Variation
GEOMETRY APPLICATIONS
GEOMETRY APPLICATIONS Chapter 3: Parallel & Perpendicular Lines Name: Teacher: Pd: 0 Table of Contents DAY 1: (Ch. 3-1 & 3-2) SWBAT: Identify parallel, perpendicular, and skew lines. Identify the angles
Geometry Unit 5 Geometric and Algebraic Connections. Table of Contents
Geometry Unit 5 Geometric and Algebraic Connections Table of Contents Lesson 5 1 Lesson 5 2 Distance.p. 2-3 Midpoint p. 3-4 Partitioning a Directed Line. p. 5-6 Slope. p.7-8 Lesson 5 3 Revisit: Graphing