# Chapter 8 Solving Systems of Linear Equations Graphically

Size: px
Start display at page:

Transcription

1 Name: Chapter 8 Solving Systems of Linear Equations Graphically 8.1 System of Linear Equations and Graphs Outcome: 1. Interpret graphical reasoning through the study of relations 3. Demonstrate an understanding of slope with respect to: Rise and run Line segments and lines Parallel lines Perpendicular lines 9. Solve problems that involve systems of linear equations in two variables graphically Definitions: Point of Intersection: a point at which two lines touch or cross ***HAVE YOU VERIFY THE POINT IN BOTH EQUATIONS*** Determine the point of intersection on the graph. System of Linear Equations: two or more linear equations involving common variables. Example: - 2x + y = 3 and 4x - 11 = y Non-example: ½ x - y = 5 and 2s +1 = t Solution (to a system of linear equations): a point of intersection of the lines on a graph An ordered pair that satisfies both equations Verify that (1, 3) satisfies y = x + 2 and y = 2x + 1:

2 A pair of values occurring in the table of values of both equations x + y = 5 2x + y = Example 1: Graph the two linear equations, 2x + y = 2 and x - y = 7, on paper and on your calculator. a) Identify the point of intersection b) Verify the solution

3 Example 2: a) Consider the system of linear equations 3x + y = 4 and x - y = 0. Identify the point of intersection of the lines by graphing. Verify the solution. Example 3: Davidee earns \$40 plus \$10 per hour. Carmen earns \$50 plus \$8 per hour. a) Represent the linear system relating the earnings numerically and graphically. b) Identify the solution to the linear system and explain what it represents.

4 Example 4: For each system of linear equations, verify whether the given point is a solution. a) x + 4y = 22 3x y = 2 (2, 5) b) 2 x + 3y = 12 4x 3y = 6 (-3, -2) Example 5: Eric works on the 23rd floor of a building. It takes Eric 90s to walk down the stairs to the 14th floor. Nathan works on the 14th floor and needs to go up to the 30th floor. He knows it will take 40s by elevator if the elevator makes no other stops. Suppose both men leave their offices at the same time. Create a graph to model their travel. b) What does the point of intersection represent?

5 Key Ideas Systems of linear equations can be modelled numerically, graphically, or algebraically. The solution to a linear system is a pair of values that occurs in each table of values, intersection point of the lines, or an ordered pair that satisfies each equation. A solution to a system of linear equations can be verified using several methods: Substitute the value for each variable and evaluate the equation Create a graph and identify the point of intersection Create table of values and identify the pair of values that occurs in each table Textbook Questions: Pg. 427 #1-15

6 8.2 Modelling and Solving Linear Systems Outcome: 1. Interpret graphical reasoning through the study of relations 7. Determine the equations of a linear relation, given: A graph A point and the slope Two points A point and the equation of a parallel or perpendicular line to solve problems. 9. Solve problems that involve systems of linear equations in two variables graphically Example 1: Translate the followings words or phrases into mathematical language. a) 7 times as many years increased by 3 b) 3 less than a number 3 decreased by a number c) Option A charges a one-time \$30 fee and then \$8 per hour Option B charges \$14 per hour Example 2: During a performance by a theatre company, the main act was on stage for 3 min less than twice the time of the opening act. Together, the two acts performed for 132 min. a) Write a system of linear equations to represent the length of time each act performed.

7 b) What is the solution to their linear system? What does the solution represent? Example 3: Two pools start draining at the same time. The larger pool contains L of water and rains at a rate of 25L/min. The smaller pool contains L of water and drains at a rate of 10L/min. a) Model the draining of the pools using a system of linear equations. b) Represent the linear system graphically on your calculator. Describe how the information shown in the graph relates to the pools. Example 4: Jamie is travelling with her family from Castlegar, BC, to Pincher Creek, AB. Her dad and cousin do all of the driving. The 440-km trip takes 5.25 hours, excluding stops. Jamie s father drives at an average speed of 90 km/hr. Her cousin drives at 80km/hr. a) What system of linear equations could help Jamie determine the length of time each person drove? b) How many hours does each person drive?

8 Key Ideas When modelling word problems, assign variables that are meaningful to the context of the problem. To assist in visualizing or organizing word problem, you can use a diagram and/or a table of values If a situation involves quantities that change at constant rates, you can represent it using a system of linear equations. If you know the initial value and rates, you can write the equations directly in slope-intercept form because the initial value is the y-intercept and the rate is the slope. Otherwise, you can determine the rate of change using start and end values. Textbook Questions: Pg. 440 #1-8, 10-11, 13

9 8.3 Number of Solutions for Systems of Linear Equations Outcome: 1. Interpret graphical reasoning through the study of relations 9. Solve problems that involve systems of linear equations in two variables graphically Definitions: Coincident Lines: lines that occupy the same position. In a graph of two coincident lines, any point of either line lies on the other line. Example 1: Four runners travel on a long, straight stretch of the Trans-Canada Highway. Their current distances and speeds are shown in the table of values. Current Distance (km) Current Speed (km/h) Sarah 4 9 Steve Lucy 3 11 Mark 4 9 For each pair of vehicles below, Write a system of linear equations representing their travel from this point forward Graph each system of linear equations Identify and interpret the solution to each linear system a) Sarah and Steve

10 b) Sarah and Mark c) Lucy and Mark

11 Example 2: Predict the number of solutions for each system of linear equations. Explain your reasoning, and then confirm each answer by graphing the linear system. a) x + 2y = 4 y = -½ x + 4 b) 6y - 4x = 6 y = ⅔ x + 4 c) y = 3x - 1 y = 2x - 1

12 Example 3: Determine, by inspection, whether each linear system has an infinite number of solutions or no solution. Explain your reasoning. a) 2x + 10y - 16 = 0 x + 5y - 8 = 0 b) x + 2y + 4 = 0 x + 2y - 6 = 0

13 Key Ideas A system of linear equations can have one solution, no solutions, or an infinite number of solutions. Before solving, you can predict the number of solutions for a linear system by comparing the slopes and y-intercepts of the equations. Intersecting Lines Parallel Lines Coincident Lines One solution No solutions Infinite solutions Different slopes Same slope Same slope Y-intercepts can be the same or different Different y-intercepts Same y-intercept For some linear systems, reducing the equations to lowest terms and comparing the coefficients of the x-terms, y-terms, and constants may help you predict the number of solutions. Textbook Questions: Pg. 454 # 1-9, 11-15

### Note: Two perpendicular lines form a system of the first type. (Nothing special about being )

Math 100 Elementary Algebra Sec 7.1: Solving Linear Systems by Graphing Two or more equations [inequalities] in several variables that are considered simultaneously are called a system of equations [inequalities].

### Math 10C: Systems of Equations PRACTICE EXAM

Math 10C: Systems of Equations PRACTICE EXAM 1. An online music store offers two payment methods: 1) The customer pays a monthly subscription fee of 8.00 and songs can be downloaded for 0.70 each. 2) The

### Math 0312 Intermediate Algebra Chapter 1 and 2 Test Review

Math 0312 Intermediate Algebra Chapter 1 and 2 Test Review Name Date MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the equation for y. 1)

### average rate of change

average rate of change Module 2 : Investigation 5 MAT 170 Precalculus August 31, 2016 question 1 A car is driving away from a crosswalk. The distance d (in feet) of the car from the crosswalk t seconds

### Exam 1 Review Sp16 O Brien. Exam 1 Review:

Exam Review:..6 Directions: Try to work the following problems with your book, notes, and homework closed. You may have your graphing calculator and some blank paper. The idea is to practice working problems

### MATHEMATICAL METHODS UNIT 1 Chapter 1 Reviewing Linear Equations Chapter 2 Coordinate geometry & linear relations

REVIEWING LINEAR EQUATIONS E da = q ε ( B da = 0 E ds = dφ. B ds = μ ( i + μ ( ε ( dφ 3 MATHEMATICAL METHODS UNIT 1 Chapter 1 Reviewing Linear Equations Chapter 2 Coordinate geometry & linear relations

### MAT 1033 Final Review for Intermediate Algebra (Revised April 2013)

1 This review corresponds to the Charles McKeague textbook. Answers will be posted separately. Section 2.1: Solve a Linear Equation in One Variable 1. Solve: " = " 2. Solve: "# = " 3. Solve: " " = " Section

### Solving Problems In Physics

Solving Problems In Physics 1. Read the problem carefully. 2. Identify what is given. 3. Identify the unknown. 4. Find a useable equation and solve for the unknown quantity. 5. Substitute the given quantities.

### State Mu Alpha Theta Contest 2007 Algebra 3&4 Class Test

State Mu Alpha Theta Contest 00 Algebra & Class Test. Rationalize the denominator: + B. + C. +. On a recent trip, Ellie drove km in the same length of time Carol took to drive 98 km. Ellie s speed was

### Extra Practice Chapter 5. Topics Include: Direct & Partial Variation Slope Slope as a Rate of Change First Differences

Extra Practice Chapter 5 Topics Include: Direct & Partial Variation Slope Slope as a Rate of Change First Differences 5.1 - Practice: Direct Variation 1. Find the constant of variation for each direct

### GUIDED NOTES 2.2 LINEAR EQUATIONS IN ONE VARIABLE

GUIDED NOTES 2.2 LINEAR EQUATIONS IN ONE VARIABLE LEARNING OBJECTIVES In this section, you will: Solve equations in one variable algebraically. Solve a rational equation. Find a linear equation. Given

### Pair of Linear Equations in Two Variables

Pair of Linear Equations in Two Variables Linear equation in two variables x and y is of the form ax + by + c= 0, where a, b, and c are real numbers, such that both a and b are not zero. Example: 6x +

### Elimination Exploring Linear Systems QUIZ ( ) Solving Problems with Systems of Equations. Distance/Velocity/Time Problems WS 1.

UNIT 1 SYSTEMS OF LINEAR EQUATIONS Lesson TOPIC Homework Sept. 4 1.0 Sept. 5 1.1 1.1 Sept. 6 1.2 1.3 Sept. 7 1.3 1.4 Sept. 10 Sept. 11 Sept. 12 Sept. 13 Sept. 14 Sept. 17 Sept. 18 Sept. 20 1.4 1.6 1.5

### Unit 3 Review for SchoolNet Test

Unit 3 Review for SchoolNet Test Student Class Date 1. The table below lists pairs of x- and y-coordinates that represent points on the graph of a linear equation. x y 3 10 5 5 7 0?? 11 10 Which coordinates

### AP Physics 1: Summer Assignment

AP Physics 1: Summer Assignment- Part 1 AP Physics 1: Summer Assignment Welcome to AP Physics 1! Attached is the summer assignment, which consists of two parts. The first part is an easy read about the

### I. ORDER OF OPERATIONS

ALGEBRA II HONORS REVIEW PACKET NAME This packet contains all of the material that you should have mastered in Algebra I. You are responsible for reviewing this material over the summer and expect an assessment

### 7 = 8 (Type a simplified fraction.)

Student: Date: Assignment: Exponential and Radical Equations 1. Perform the indicated computation. Write the answer in scientific notation. 3. 10 6 10. 3. 4. 3. 10 6 10 = (Use the multiplication symbol

### Speed ( v ) is the distance an object travels during a given time interval divided by the time interval.

v 8.2 Average Velocity Speed ( v ) is the distance an object travels during a given time interval divided by the time interval. Speed is a scalar quantity. The SI unit for speed is metres per second (m/s).

### Algebra I Study Topics

This Algebra I Study Guide contains clear, straight-forward problems that represent the topics covered in a complete Algebra I course. After completing the study guide without a calculator, correct it

### For any negative real number x, the statement

1. Which equation does not have a solution? w + 3 = 3w + w + 3 = w + 5 w + 3 = 4w + 6 w + 3 = w + w + 3. Which expression represents the phrase the quotient of three less than four times a number n and

### Math 10 Lesson 5-1 System of Linear Equations Graphical solution

Math 10 Lesson 5-1 System of Linear Equations Graphical solution I. Lesson Objectives: 1) Use the graphs of the equations of a linear system to estimate its solution. 2) Learn how to write a system of

### 6-1 Using Graphs to Solve a System of Equations., together form a system of linear equations.

AH Notes# 6-1 Using Graphs to Solve a System of Equations Two or more linear equations in the same variables, such as, together form a system of linear equations. The solution to the system is any/all

### Midterm: Wednesday, January 23 rd at 8AM Midterm Review

Name: Algebra 1 CC Period: Midterm: Wednesday, January 23 rd at 8AM Midterm Review Unit 1: Building Blocks of Algebra Number Properties (Distributive, Commutative, Associative, Additive, Multiplicative)

### Mathematics 10C. UNIT FIVE Linear Functions. Unit. Student Workbook. y 2. - y 1 x 2. rise. m = - x 1 run. y = mx + b. m = m original. 1 moriginal.

Mathematics 10C Student Workbook Unit m = y 2 - y 1 x 2 - x 1 run rise Lesson 1: Slope of a Line Approximate Completion Time: 2 Days y = mx + b Lesson 2: Slope-Intercept Form Approximate Completion Time:

### Position, Speed and Velocity Position is a variable that gives your location relative to an origin. The origin is the place where position equals 0.

Position, Speed and Velocity Position is a variable that gives your location relative to an origin. The origin is the place where position equals 0. The position of this car at 50 cm describes where the

### MATH 126 TEST 1 SAMPLE

NAME: / 60 = % MATH 16 TEST 1 SAMPLE NOTE: The actual exam will only have 13 questions. The different parts of each question (part A, B, etc.) are variations. Know how to do all the variations on this

### Section 1 Force and Motion: Review

Section 1 Force and Motion: Review 1. MAIN IDEA How does a motion diagram represent an object s motion? A motion diagram shows the positions of a moving object at equal time intervals. 2. Motion Diagram

### Algebra 1 Fall Review

Name Algebra 1 Fall Review 2013-2014 Date 1. Write an inequality to best represent the graph shown at right. (A.1.D.) m: b: inequality: 2. Write an inequality to best describe the graph shown at right.

### 8.3. Number of Solutions for Systems of Linear Equations. Investigate Number of Solutions for Systems of Linear Equations.

8.3 Number of Solutions for Sstems of Linear Equations Focus on explaining wh sstems of linear equations can have different numbers of solutions identifing how man solutions a sstem of linear equations

### MATH ALGEBRA AND FUNCTIONS

Students: 1. Students express quantitative relationships using algebraic terminology, expressions, equations, inequalities and their graphs. 1. Use variables and appropriate operations to write an expression,

### Equations and Inequalities in One Variable

Name Date lass Equations and Inequalities in One Variable. Which of the following is ( r ) 5 + + s evaluated for r = 8 and s =? A 3 B 50 58. Solve 3x 9= for x. A B 7 3. What is the best first step for

### Moving Straight Ahead - Unit Test Review Sheet

Name: Class: Date: ID: A Moving Straight Ahead - Unit Test Review Sheet Short Answer 1. Brent's Video Shack charges \$1.50 to rent a video game for a night. Mr. Buck's Entertainments opens a new store in

### East Greenwich Mathematics Summer Review Material for Students Entering Algebra I, Part II Directions:

East Greenwich Mathematics Summer Review Material for Students Entering Algebra I, Part II 2016-2017 Directions: In order to earn full credit, show all work in the spaces provided. Do all work without

### 5) A stone is thrown straight up. What is its acceleration on the way up? 6) A stone is thrown straight up. What is its acceleration on the way down?

5) A stone is thrown straight up. What is its acceleration on the way up? Answer: 9.8 m/s 2 downward 6) A stone is thrown straight up. What is its acceleration on the way down? Answer: 9.8 m/ s 2 downward

### Precalculus: Linear Equations Practice Problems. Questions. 1. Solve for x when 2 3 x = 1 15 x Solve for x when x 2 + x 5 = 7 10.

Questions. Solve for x when 3 x = 5 x + 3 5.. Solve for x when x + x 5 = 7 0. 3. Solve for x when 0 3 x = x. 4. Is 4 a solution to (y ) + = 3 (3y 4)? 8 5. Solve for x when 4 5 x 3 = 3x +. 6. Solve for

1. C 2. x 36 3. r 12. x 3 5. 17 x 1.16666 12 6. x 6 7. 8. 9. 2x3 5x6 2(3) 3 9 3x 9 or 5(3) 6 9 x 3 Student should grid 9 12 2L W 2 or W 6L or W L 6 A C H B 10. x 11. x 1 12. x 11 11 1 13. 20 x 30 1. 15.

### 5.2.3 Systems of Equations (Substitution Method)

Lesson: 5.2.3 Systems of Equations (Substitution Method) 5.2.3 Alternate Systems of Equations (Substitution Method) Teacher Lesson Plan CC Standards 8.EE.8b 8.EE.8c Solve systems of two linear equations

Grade 8 Curriculum Map 2007-2008 Moving Straight Ahead 25 Days Curriculum Map 2007-2008 CMP2 Investigations Notes Standards 1.2 Finding and Using Rates Walking Rates and Linear Relationships 1.3 Raising

### Physics 12. Chapter 1: Vector Analysis in Two Dimensions

Physics 12 Chapter 1: Vector Analysis in Two Dimensions 1. Definitions When studying mechanics in Physics 11, we have realized that there are two major types of quantities that we can measure for the systems

### (A) 20% (B) 25% (C) 30% (D) % (E) 50%

ACT 2017 Name Date 1. The population of Green Valley, the largest suburb of Happyville, is 50% of the rest of the population of Happyville. The population of Green Valley is what percent of the entire

### Kinematics 7 Solutions. 7.1 Represent and Reason a) The bike is moving at a constant velocity of 4 m/s towards the east

Kinematics 7 Solutions 7.1 Represent and Reason a) The bike is moving at a constant velocity of 4 m/s towards the east b) For the same motion, a position versus time graph would be a straight line at a

### Graphical Solutions of Linear Systems

Graphical Solutions of Linear Systems Consistent System (At least one solution) Inconsistent System (No Solution) Independent (One solution) Dependent (Infinite many solutions) Parallel Lines Equations

### NAME DATE PER. Review #11 Solving Systems of Equations 1. Write the linear function that includes the points (4, 9) and (-2, -6).

Review #11 NAME DATE PER. Review #11 Solving Systems of Equations 1. Write the linear function that includes the points (4, 9) and (-2, -6). 2. The graph of a line that contains the points (-3, 2) and

### 4 The Cartesian Coordinate System- Pictures of Equations

4 The Cartesian Coordinate System- Pictures of Equations Concepts: The Cartesian Coordinate System Graphs of Equations in Two Variables x-intercepts and y-intercepts Distance in Two Dimensions and the

### Correlation of From Patterns to Algebra to the Ontario Mathematics Curriculum (Grade 4)

Correlation of From Patterns to Algebra to the Ontario Mathematics Curriculum (Grade 4) Strand: Patterning and Algebra Ontario Curriculum Outcomes By the end of Grade 4, students will: describe, extend,

### Rational Numbers, Irrational Numbers, Whole Numbers, Integers, Natural Numbers

College Readiness Math Semester Eam Review Write the set using interval notation and graph the inequality on the number line. 1) { -6 < < 9} Use the order of operations to simplify the epression. 4 (8

### 4. The table shows the number of toll booths driven through compared to the cost of using a Toll Tag.

ALGEBRA 1 Fall 2016 Semester Exam Review Name 1. According to the data shown below, which would be the best prediction of the average cost of a -bedroom house in Georgetown in the year 2018? Year Average

### Displacement, Velocity & Acceleration

Displacement, Velocity & Acceleration Honors/AP Physics Mr. Velazquez Rm. 254 1 Velocity vs. Speed Speed and velocity can both be defined as a change in position or displacement over time. However, speed

### 3. If a coordinate is zero the point must be on an axis. If the x-coordinate is zero, where will the point be?

Chapter 2: Equations and Inequalities Section 1: The Rectangular Coordinate Systems and Graphs 1. Cartesian Coordinate System. 2. Plot the points ( 3, 5), (4, 3), (3, 4), ( 3, 0) 3. If a coordinate is

### SCIENCE 1206 Unit 3. Physical Science Motion

SCIENCE 1206 Unit 3 Physical Science Motion Section 1: Units, Measurements and Error What is Physics? Physics is the study of motion, matter, energy, and force. Qualitative and Quantitative Descriptions

### Chapter 9 Solving Systems of Linear Equations Algebraically

Name: Chapter 9 Solving Systems of Linear Equations Algebraically 9.1 Solving Systems of Linear Equations by Substitution Outcomes: 1. Interpret algebraic reasoning through the study of relations 9. Solve

### Algebra II Vocabulary Word Wall Cards

Algebra II Vocabulary Word Wall Cards Mathematics vocabulary word wall cards provide a display of mathematics content words and associated visual cues to assist in vocabulary development. The cards should

### Graphing and Writing Linear Equations Review 3.1, 3.3, & 4.4. Name: Date: Period:

Graphing and Writing Linear Equations Review.1,., 4.1-4. & 4.4 Algebra I Name: Date: Period: Quest Topics Section.1 linear versus nonlinear rewrite linear equations in standard form: Ax By C find and use

### 6-4 Solving Special Systems

Warm Up Solve each equation. 1. 2x + 3 = 2x + 4 2. 2(x + 1) = 2x + 2 3. Solve 2y 6x = 10 for y Solve by using any method. 4. y = 3x + 2 2x + y = 7 5. x y = 8 x + y = 4 Know: Solve special systems of linear

### Skills Practice Skills Practice for Lesson 1.1

Skills Practice Skills Practice for Lesson. Name Date Tanks a Lot Introduction to Linear Functions Vocabulary Define each term in your own words.. function 2. linear function 3. independent variable 4.

1. increasing: 1.5 t 2 t.5 decreasing: 0 t 1.5 3 t constant: 2 t 3.5 t 6 2. x 36 3. r 29 3. x 3.7 5. x 6 6. Acceptable answers: 13/, 3.25 Unacceptable Grid in: 3 1/ P 2 1 W L or W P L or W P L 2 2 2 7.

### Mourning Sr. High. Algebra II Summer Assignment. 3. If you need help, there are links provided on each page with extra help resources.

Mourning Sr. High Algebra II Summer Assignment Pre AP Algebra students and Honors Algebra students must complete the entire assignment. Regular Algebra students complete the following: Fractions # s 3

### Motion Along a Straight Line

PHYS 101 Previous Exam Problems CHAPTER Motion Along a Straight Line Position & displacement Average & instantaneous velocity Average & instantaneous acceleration Constant acceleration Free fall Graphical

### Recognise the Equation of a Circle. Solve Problems about Circles Centred at O. Co-Ordinate Geometry of the Circle - Outcomes

1 Co-Ordinate Geometry of the Circle - Outcomes Recognise the equation of a circle. Solve problems about circles centred at the origin. Solve problems about circles not centred at the origin. Determine

### 5. Evaluate. 9. Evaluate each of the following: Answers. 1. a) a) 1 6

MPMD Exam Review Unit (Focus on Integers and Rationals). Add or subtract. Express your answers in lowest terms. 7 + 8 8 6 + 4 6 7. Evaluate. 6 + 4 ( ) ( 7) + ( )( ) (6 8) ( 8 + ) (4 ) ( ) e) 0 + 8 ( 4)

### 1) Find the equations of lines (in point-slope form) passing through (-1,4) having the given characteristics:

AP Calculus AB Summer Worksheet Name 10 This worksheet is due at the beginning of class on the first day of school. It will be graded on accuracy. You must show all work to earn credit. You may work together

### Unit 5: Moving Straight Ahead Name: Key

Unit 5: Moving Straight Ahead Name: Key 1.1: Finding and Using Rates 1.: Tables, Graphs and Equations 1.3: Using Linear Relationships Independent Variable: One of the two variables in a relationship. Its

### Elementary Algebra SAMPLE Final Examination Fall 2017

Elementary Algebra NAME: SAMPLE Final Examination Fall 2017 You will have 2 hours to complete this exam. You may use a calculator but must show all algebraic work in the space provided to receive full

### The total time traveled divided by the total time taken to travel it. Average speed =

Unit 3: Motion V = d t Average speed The total time traveled divided by the total time taken to travel it Mathematically: Average speed = Total Distance Travelled Total Time Traveled So just how fast were

### Chapter 2. Motion along a Straight Line

Chapter 2 Motion along a Straight Line 1 2.1 Motion Everything in the universe, from atoms to galaxies, is in motion. A first step to study motion is to consider simplified cases. In this chapter we study

### Algebra Supplement Homework Packet #1

Algebra Supplement Homework Packet #1 Day 1: Fill in each blank with one of the words or phrases listed below. Distributive Real Reciprocals Absolute value Opposite Associative Inequality Commutative Whole

### P.7 Solving Inequalities Algebraically and Graphically

54 CHAPTER P Prerequisites What you ll learn about Solving Absolute Value Inequalities Solving Quadratic Inequalities Approximating Solutions to Inequalities Projectile Motion... and why These techniques

CALCULUS 30: OUTCOME 4A DAY 1 SLOPE AND RATE OF CHANGE To review the concepts of slope and rate of change. VIDEO LINKS: a) https://goo.gl/r9fhx3 b) SLOPE OF A LINE: Is a measure of the steepness of a line

### Solving Systems of Linear Equations

Section 2.3 Solving Systems of Linear Equations TERMINOLOGY 2.3 Previously Used: Equivalent Equations Literal Equation Properties of Equations Substitution Principle Prerequisite Terms: Coordinate Axes

### Name Date Class. Original content Copyright by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.

Name Date Class 1-1 Graphing Linear Equations Write the correct answer. 1. The distance in feet traveled by a falling object is found by the formula d = 16t where d is the distance in feet and t is the

### The key difference between speed and velocity is the. object s motion, while velocity designates an object s speed plus the direction of its motion.

Article retrieved from Brittanica, Retrieved 6/27/2016 Velocity Velocity has a scientific meaning that is slightly different from that of speed. Speed is the rate of an object s motion, while velocity

### This is a review packet for the entire fall semester of Algebra I at Harrison.

HARRISON HIGH SCHOOL ALGEBRA I Fall Semester Review Packet This is a review packet for the entire fall semester of Algebra I at Harrison. You are receiving it now so that: you will have plenty of time

### 1.1 Motion and Motion Graphs

Figure 1 A highway is a good example of the physics of motion in action. kinematics the study of motion without considering the forces that produce the motion dynamics the study of the causes of motion

### 3. A beam or staircase frame from CSP costs \$2.25 for each rod, plus \$50 for shipping and handling.

Pg. 13: #3 3. A beam or staircase frame from CSP costs \$2.25 for each rod, plus \$50 for shipping and handling. a. Complete the following table to show the costs for beams of different lengths. Beam Length

### 12/06/2010. Chapter 2 Describing Motion: Kinematics in One Dimension. 2-1 Reference Frames and Displacement. 2-1 Reference Frames and Displacement

Chapter 2 Describing Motion: Kinematics in One Dimension 2-1 Reference Frames and Displacement Any measurement of position, distance, or speed must be made with respect to a reference frame. For example,

### Section 3.4 Writing the Equation of a Line

Chapter Linear Equations and Functions Section.4 Writing the Equation of a Line Writing Equations of Lines Critical to a thorough understanding of linear equations and functions is the ability to write

### Review for the Algebra EOC

Review for the Algebra EOC The test is Thursday, January 26 th, 2017 The answer key for this review booklet can be found at: www.mrshicklin.pbworks.com 1. A 1,500-gallon tank contains 200 gallons of water.

### Paper-Based: 8th Grade Comprehensive Mathematics Assessment

Student Name: Teacher: Date: District: Miami-Dade County Public Schools Assessment: 08 Mathematics Mathematics Exam 3 Description: Paper-Based: 8th Grade Comprehensive Mathematics Assessment Form: 201

### Chapter 1 Skills Points and Linear Equations

Example 1. Solve We have Chapter 1 Skills Points and Linear Equations t = 3 t t = 3 t for t. = ( t)( t) = ( t) = 3( t) = 4 4t = 6 3t = = t t = 3 ( t)( t) t Example. Solve We have = A + Bt A Bt for t. =

### Motion and Forces study Guide

Motion and Forces study Guide Completion Complete each statement. 1. The motion of an object looks different to observers in different. 2. The SI unit for measuring is the meter. 3. The direction and length

### 8.G.1. Illustrative Mathematics Unit 1. 8.G.1a 8.G.2 8.G.3. 8.G.1c. Illustrative Mathematics Unit 2 8.G.2

Illustrative Mathematics Unit 3 Illustrative Mathematics Unit 2 Illustrative Mathematics Unit 1 8.G.1 8.G.1a 8.G.1b 8.G.1c 8.G.2 8.G.3 8.G.1 8.G.2 8.G.3 8.EE.5 8.EE.6 1st Nine Weeks Rigid Transformations

### AP Physics 1 Summer Assignment 2018 Mrs. DeMaio

AP Physics 1 Summer Assignment 2018 Mrs. DeMaio demaiod@middletownk12.org Welcome to AP Physics 1 for the 2018-2019 school year. AP Physics 1 is an algebra based, introductory college-level physics course.

### Beginning Algebra. 1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions

1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions Beginning Algebra 1.3 Review of Decimal Numbers and Square Roots 1.4 Review of Percents 1.5 Real Number System 1.6 Translations:

### State the condition under which the distance covered and displacement of moving object will have the same magnitude.

Exercise CBSE-Class IX Science Motion General Instructions: (i) (ii) (iii) (iv) Question no. 1-15 are very short answer questions. These are required to be answered in one sentence each. Questions no.

### STANDARDS OF LEARNING CONTENT REVIEW NOTES. ALGEBRA I Part II 1 st Nine Weeks,

STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I Part II 1 st Nine Weeks, 2016-2017 OVERVIEW Algebra I Content Review Notes are designed by the High School Mathematics Steering Committee as a resource

### PETERS TOWNSHIP HIGH SCHOOL

PETERS TOWNSHIP HIGH SCHOOL COURSE SYLLABUS: ALGEBRA 1 ACADEMIC Course Overview and Essential Skills This course is a study of the language, concepts, and techniques of Algebra that will prepare students

### Final Review ( Warm Up ) a) 16 in = cm b) 5 yd = m c) 6 mi = km. d) 19 m = ft e) 150 km = mi f) 10 lb = kg. 1.7 m 100 cm 6in. 2.

Math 10 Final Review ( Warm Up ) Name Date: Blk Chp 1. Conversions between Metric and Imperial Units 1. Convert each measurement to the nearest tenth. a) 16 in = cm b) 5 yd = m c) 6 mi = km d) 19 m = ft

1st Nine Weeks Equations 8.EE.7 Solve linear equations in one variable including rational coefficients. a. Give examples of linear equations in one variable with one solution, infinitely many solutions,

### Math 20-1 Functions and Equations Multiple Choice Questions

Math 0- Functions and Equations Multiple Choice Questions 8 simplifies to: A. 9 B. 0 C. 90 ( )( ) simplifies to: A. B. C. 8 A. 9 B. C. simplifies to: The area of the shaded region below is: 0 0 A. B. 0

### ALGEBRA 1 CST Questions (2009)

1 Is the equation 3(x ) = 18 equivalent to 6x 1 = 18? Yes, the equations are equivalent by the ssociative Property of Multiplication. Yes, the equations are equivalent by the ommutative Property of Multiplication.

### Name: Class: Date: ID: A

Name: Class: Date: 8th Grade Advanced Topic III, Linear Equations and Systems of Linear Equations, MA.8.A.1.1, MA.8.1.1.2, MA.8.A.1.3, *MA.8.A.1.4, MA.8.A.1.5, MA.8.A.1.6 Multiple Choice Identify the choice

### Name: Class: Date: ID: A. c. the quotient of z and 28 z divided by 28 b. z subtracted from 28 z less than 28

Name: Class: Date: ID: A Review for Final Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Give two ways to write the algebraic expression z 28 in words.

### Chapter 2 1D KINEMATICS

Chapter 2 1D KINEMATICS The motion of an American kestrel through the air can be described by the bird s displacement, speed, velocity, and acceleration. When it flies in a straight line without any change

### INTERMEDIATE ALGEBRA TEST ONE PRACTICE SOLUTIONS SHOW ALL CALCULATIONS AND SIMPLIFY ANSWERS. PAGE 1 OF 8.

SHOW ALL CALCULATIONS AND SIMPLIFY ANSWERS. PAGE 1 OF 8. 1. (a) Solve the inequality x 3 7. 7 x 3 7, 4 x 10 [ 4, 10] 7 + 3 x 3 + 3 7 + 3 or (b) Graph the solution to the inequality in (a), above. -6-4

### (MATH 1203, 1204, 1204R)

College Algebra (MATH 1203, 1204, 1204R) Departmental Review Problems For all questions that ask for an approximate answer, round to two decimal places (unless otherwise specified). The most closely related

FLORIDA STANDARDS TO BOOK CORRELATION FOR GRADE 7 ADVANCED After a standard is introduced, it is revisited many times in subsequent activities, lessons, and exercises. Domain: The Number System 8.NS.1.1

### b. Write an equation that represents the revenue (income). c. A set of two (or more) linear equations is called a system of linear

How can the given information be represented with equations? What is a solution to a two-variable equation? How can this problem be represented on a graph? How does the solution appear on the graph? Your

### The steps in Raya s solution to 2.5 (6.25x + 0.5) = 11 are shown. Select the correct reason for line 4 of Raya s solution.

A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear functions. Unit 2: Reasoning with Linear Equations and Inequalities The perimeter