Algebra 2 Honors Unit 1 Review of Algebra 1
|
|
- Nathan Waters
- 5 years ago
- Views:
Transcription
1 Algebra Honors Unit Review of Algebra Day Combining Like Terms and Distributive Property Objectives: SWBAT evaluate and simplify expressions involving real numbers. SWBAT evaluate exponents SWBAT combine like terms SWBAT plug values into expressions SWBAT apply Distributive and Double Distributive Properties Exponent Vocabulary Base The number that will be multiplied Coefficient the number in front of an exponent (assume it s a if there is none). Exponent the number of times the base will be multiplied Examples Simplify the following expressions.. 4. ( ) 4. ( 5) 8 ( ) ( ) ( ) ( ) 6 ( 5)( 5)( 5) x 6. (y) x ( y)( y)( y) 8y Adding and Subtracting Expressions Vocabulary Terms Like Terms A part or piece of an expression or equation separated by a + or Terms with the exact variable makeup Constant Simplified Number with no variable attached Combining all like terms or solving
2 Distributive Property Double Distributive Property (FOIL) a c d a( b c) ab ac b ac ad bc bd Examples Simplify the following expressions. 7. (7z +z) + (6z +z) 8. (9v 8w) (0w 5v) 7z z 6z z 7z 6z z z z 5 z 9v 8w 0w 5v 9v 8w 0w 5v 9v 5v 8w 0w 4v8w 9. x7x 4 0. x x x x 5 x (7x 4) 7 x 4 x x(x x x 5) x x x x. (x 9)(x + 4). (x + 4)(x + 6) 6x 8x 7x 6 6x 9x6 4 x 8x 4x 4 4 x x 4
3 Imposters of Double Distributive For the following examples, explain why they are not a double distributive, and then simplify.. (x 5y) (4x + y) 4. (4m ) There is a +/- sign in-between the ( ) There is only one set of ( ) So it is not double distributive, it is so that means this is just Combining like terms distributive property x 4x 5y y x 8y (4m ) 8m 4 8m 7 5. (x )(5x x + ) 6. m (x )(5x x ) 0x x 6x 5x x 9 0x 7x 9x 9 mmm m m m 4 m m m m m 4m 4 m m 4m 4m m m m m m m m m 6 8
4 Day Solve Linear Equations and Inequalities Objectives: SWBAT solve linear equations and inequalities. SWBAT use a formula. SWBAT isolate any variable in a formula or literal equation. GOLDEN RULE OF SOLVING LINEAR EQUATIONS Six Steps to help solve equations.. PEMDAS. Combine like terms that are on the same side of the = sign. If there are variables on each side of the = sign, find the smallest variable and move it to the other side. 4. Solve the two step equation by isolating the variable term 5. Multiply/Divide the coefficient of the variable so that there is a single positive variable = a number 6. Check your work Examples Solve the following equations. x. 8 7 x x 5 7 x x w. 5 m 6 w w w 5 0 w 5 m 6 5 m m 6 m m m
5 4. n 6 n x x 7 4x x7 n 6 n4 n n 6 n n 6. x 4 6x 6 7. x 4 x 4x 0 7x x 7 7x 7 x x (x) 4 6x 6 6x 4 6x 6 6x 6 6x 6 6 x 6x 6 6 All Real Numbers ( x 6) 4 7 x x 4 7 x x6 7 x x x 6 7 No Solution Formulas and Rewriting Equations 5 Example: The equation C F is used to change temperature from Fahrenheit to 9 Celsius. Find the equivalent Celsius temperature for each of the following Fahrenheit temperatures. 8. F C 5 C 9 5 C 80 9 Examples C 00 Solve each equation for the indicated variable A b b h, for b A b b h A b b h 0. a + b = c, for a. a b c b b a c b a c b a c b 5 70 F F F h A b b h b b A b b h F 58
6 Day Slope Intercept Form and Writing Equations of Lines Objectives: SWBAT graph lines SWBAT write the equation of lines from their graphs SWBAT write the equation of lines from a point and a slope Slope rise y y m run x x Slope intercept form y mx b m slope b y intercept y mx b begin with move with " b" " m" Examples State the slope and the y-intercept for each of the following equations.. y = x. x y. y = 5 4. x = m b m b 0 m 0 or no slope b 5 m undefined b doesn ' t have one Rewrite each equation into slope-intercept form. 5. x + y = 5 6. x y = 0 xy 5 x x y x5 5 y x x y 0 x x y x y x
7 Graph each equation. 7. f ( x) x 8. x y = 4 m or b Solve for " y" first y x m b Put a dot on - on y axis, Put a dot on - on y axis, then move up right then move up right 9. y 0. x 5 m 0 or no slope b m undefined b doesn ' t have one Horizontal Lines: y = any number Vertical Lines: x = any number
8 Write the equation of each line graphed below. a c d b e Line a: yx Line d: y 7 Line b: y x Line e: y x 6 Line c: x 9 Point Slope form of a linear equation Traditional Formula (H, K) y y m x x m slope x x y y x, y a point y a x h k a slope x x y y hk, a point Use point slope form to write an equation for each problem. Then change it to slope intercept form. Pick either formula to use they both will work. It passes through (4, ) with a slope of 4 y 4 x 4 y 4x6 y 4x8 y 4 x 4 y 4x6 y 4x 8
9 . Contains (4,) and the slope is undefined Since the slope is undefined means you will have a vertical line. x =. Contains the origin and ( 6, ) You must find the slope between those two points y y 0 m x x You can now use either point so use (0,0) because it is the easiest y 0 x 0 y 0 x 0 y x y x 0 0 y x00 y x 4. Contains an x-intercept of and y-intercept of You must find the slope between those two points (,0) and (0,) y y 0 m x x 0 You can now use either point so use (0,) because it is the easiest y x 0 y x y x y x 0 y x 0 y x
10 Day 4 Solving Systems of Equations by Graphing and Substitution Objectives: SWBAT solve systems of equations by graphing. Objectives: SWBAT solve systems of equations by Substitution What is a system of linear equations? Two or more equations with the same variable make-up What is a solution to a system of equations? A Point Finding Solutions to Systems of Equations by Graphing ) Graph Both Lines ) Find where they intersect ) y = x + and y = x y x m b y x m b Intersection point Check, y x y x
11 Introduction to Substitution Take the value of one variable and plug it into the second equation. Find the solution to the system. x 6 y x. x 6 y 8 y x is already solved for, so plug it into the other equation. yx y 6 y y 9 y is already solved for, so plug it into the other equation. x6y8 x 6 8 x 8 x 0 x 0 6,9 0,
12 Solving Systems of Equations by Substitution ) Look at the system of equations. Decide which problem will be the easiest to isolate a variable. You have 4 ways to proceed WORKER SMARTER NOT HARDER ) Decide which variable you want to solve for, and isolate that variable. ) Plug or Substitute the equation into the variable into the other equation, solve for the single variable. 4) Substitute the value of the first variable, and plug it back into the first equation. Solve for the Second equation. This will produce the part of your answer. 5) Check it! Put both values into both equations to see if it checks out. 6) ***** If the two equations are the same, the answer is infinite solutions. 7)****** If the two equations are parallel, then there is no solution. ***** If your Math is good, it does not matter where you start substituting! The answer will be the same******* Examples Solve the following systems of equations using substitution.. x = y 4 and 8x y = 6 ) Let s look at the two equations, which equation would be easier to isolate a particular variable? (EQN since it is already set equal to x ) Take the y-4, and plug into the equation 8x -y =6. It should now read, 8(y+4) - y =6 ) Solve for y. Then take that y and plug the value of y into x=y-4 4) Solve for x. 5) Check it. 8( y 4) y6 x (8) 4 0 (8) 4 4y y6 y 6 y 68 y 8 x 4 4 x (0) 0 (8)
13 4. x y = 4 and x y = 4 xy4 xy4 ( y4) y4 6y 40 y4 5y y 60 y 5 x ( 5) 4 x 04 4 x 0 (0) ( 5) ( 5) x + 6y = 9 and x 5y = 55 x 6y 9 x 6y 9 x y ( y) 5y 55 4y 6 5y 55 9y y 8 y 9 x 6( 9) 9 (5) 6( 9) 9 x x x 5 (5) 5( 9)
14 Day 5 Solving Systems of Equations by Elimination Solving Systems of Equations by Elimination Objectives: SWBAT solve systems of equations by Elimination ) Put both equations into the Standard_ form. Usually this will be Ax + By = C Form (X AND Y ON THE SAME SIDE). ) Get Matching coefficients in front of either the x or y variables. ) Add the two equations together. Make sure this cancel out one of our variables. Solve the remaining equation. You just found _First part of your answer. 4) Substitute the value for the solved variable back into either equation. Then solve the remaining Equation. This is the other Part of your answer. 5) Write your answer as a X=, and Y=. Or as an ordered pair. 6) If both variables cancel out, then if it is a true statement then it is all real numbers, and if it is a false statement then it is no solution. 7) Check your Work. Examples Solve the following systems of equations using elimination.. x y 7 and 5x y 7. 7x y and 7x y 0 xy7 5() y 7 7x y ( ) 5x y 7 5 y 7 ( ) 7x y0 8x 4 y 8 4y x y 4 y 7x 0 7x 7x x
15 . 6y + 7 = 5x and x 9 = 4y x 4y = 8 and 9x y = 54 6y7 5x 7 7 6y 5x7 5 x 5x 6y 5x 7 6y5x7 5 4y x 8y5x 0y5x5 y 4 y x 9 4y4 9 9 x4y 4 y 4y x4y 6 7 5x 7 5x 5 5x x x 4y 8 9x y 54 9x y 54 9xy 54 0x0y 0 Infinately Many Solutions 5. 8x + 5 = y and + y = 6x 8x 5 y 8 x 8x 5 y8x y 6x y y 6x y 8x y 5 6x y 8x y 5 6x y 6xy 0 6x y 0x 0y No Solution
16 Day 6 Solving Systems of Equations with Variables Objectives: SWBAT to solve a system involving three equations and three unknowns. Ordered Triple - THE ELIMINATION METHOD FOR A THREE-VARIABLE SYSTEM ) Write two Elimination Equations, (top two and bottom two but you can pick any two) ) Eliminate a variable the first equation (you should be left with an equation with two variables). ) Eliminate the same variable in the second set of equations (you should be left with an equation with two variables). 4) Line up the equations with only two variables, and perform Elimination again. Then solve that equation (you will have one part of your ordered triple). 5) Back substitute that solution into one of the equations with two variables to get the second answer (you will get the second part of your ordered triple). 6) Take those two answers, and back substitute into one of the original equations for your third answer. 7) Write you answer as an ordered triple and Check your work. Examples on the next page.
17 Examples: Solve the System.. 4x y z. x y 5z 4 6x y 4z x y z 0 6x y z x 4y z 7 Write two Elimination Equations, (top two and bottom two) 4x y z x y 5z 4 x y 5z 4 6x y 4z x y z 0 6x y z 6x y z x 4y z 7 Eliminate a variable the first equation (you should be left with an equation with two variables). 4x y z x y z 0 x y 5z 4 6x y z 4x y z 4x 6y 0z 8 8y7z9 Eliminate the same variable in the second set of equations (you should be left with an equation with two variables). x y 5z 4 6x y 4z 6x y 4z 0 6x y z xz8 6x y z x 4y z 7 6x 9y 5z 4 6x y 4z 8yz4 x 4y z 4 x 4y z 7 x5z Line up the equations with only two variables, and perform Elimination again. Then solve that equation. xz8 8y7z9 x5z 8yz4 8y 7z 9 8yz4 8y 7z 9 8yz 4 4z 4z 4 z 5 xz8 x5z 60x5z90 9x5z 9 99x 99 99x x
18 Back substitute that solution into one of the equations with two variables to get the second answer. 8y7z9 8y 7 9 8y 9 8y 8 8y 8 8 y x5z 5z 5z 5z 0 5y 0 5 y Take those two answers, and back substitute into one of the original equations for your third answer. 4x y z 4x 4x 9 4x 7 4x 8 4x 8 x 4 6x y z y 6 6 y y 4 y 6 y 6 y Write all three as an Ordered Triple This is your answer z y x,, x z y,,
19 . 4x y 5z 9 x y 8z x y z Write two Elimination Equations, (top two and bottom two) 4x y 5z 9 x y 8z x y 8z x y z Eliminate a variable the first equation (you should be left with an equation with two variables). 4x y 5z 9 x y 8z 4x y 5z 9 9x y 4z 6 Eliminate the same variable in the second set of equations (you should be left with an equation with two variables). 5x 9z 44 x y 8z x y z x y 8z x y z x 5z 8 Line up the equations with only two variables, and perform Elimination again. Then solve that equation. 5x 9z 44 x 5z 8 5x 9z 44 5 x 5z 8 5x 9z 44 5x 5z 40 4z 4 4z 4 4 z
20 Back substitute that solution into one of the equations with two variables to get the second answer. x 5z 8 x 5 8 x 5 8 x Take those two answers, and back substitute into one of the original equations for your third answer. x y z y 6 y y 9 y 4 Write all three as an Ordered Triple This is your answer z x y 4, 4, Special Cases 5. x 4y 0z 4 6. x y 5z 4 x 4y z 5 x y z x y z 9 x 4y 4z NO SOLUTION INFINATELY MANY SOLUTIONS
21 Day 7 Domain and Range Objectives: SWBAT state the domain and range from a set of points. SWBAT state the domain and range from a graph. Vocabulary ~ Domain~ The set of all inputs or x-coordinates of a function and/or graph Range~ The set of all outputs or y-coordinates of a function and/or graph Find the domain of the following sets of points Domain x coordinates inputs. (, 5), (5, 6), ( 0, 5), (, 0). ( 5, 7), (, ), (, ), (5, 5) Domain: 0,,, 5 Range: 6, 5, 0, 5 Domain: 5,,, 5 Range: 5,, 7 Find the domain and range of each function Domain: -9, -7, -5, -, - Range: -0, -6, -,, 6 domain: 9 x 9 Range: y 9
22 Domain: 4x 5 Range: 5 y 8 Domain: 8 x 0 Range: 8 y 8 Domain: 7 x 7 Range: 0 y 7 Domain: 4 x 5 Range: 6 y 6 Find the domain and range of each function Domain: x Range: y 0 Domain: x 8 Range: y
23 Domain: ARN Range: y Domain: x 0 Range: ARN Domain: ARN Range: ARN Domain: ARN Range: y 9
24 Open Interval Closed Interval Infinite Interval (Positive) Infinite Interval (Negative) Day 8 Notation Objective: SWBAT write domain and ranges in set, interval, and inequality notation Words A set of numbers greater than a and less than b A set of numbers greater than or equal to a and less than or equal to b A set of numbers where x is greater than a A set of numbers where x is less than or equal to b Inequality Notation Set Notation Interval Notation a < x < b {x a < x < b} (a, b) a x b {x a x b} [a, b] x > a x < b {x x > a} {x x a} {x x < b} {x x b} (a, + ) [a, + ) (, b) (, b] Graph Given the following graph, write the domain and range in inequality, set, and interval notation. Also, describe using words.. D Words Inequality Set Interval The domain is in-between -9 and 9 inclusive 9 x 9 {x 9 x 9} [ 9, 9] R The range is in-between and 9 inclusive y 9 {y y 9} [, 9]
25 D Words Inequality Set Interval The domain is greater than x > {x x > } (, ) R The range is less than - y < {y y < } (, ) D R Words Inequality Set Interval The domain is greater than or equal to - x {x x } [, ) The range is greater than or equal to y {y y > } [, ).
26 Words Inequality Set Interval D The domain is all real numbers < x < {x < x < } (, ) R The range is less than or equal to y {y y } (, ]
27 Day 9 Piecewise Functions Day Objectives: SWBAT graph functions with restrictive domains SWBAT evaluate piecewise functions Piecewise Function Functions or graphs that are formed by piecing together two or more functions onto one coordinate plane. Close Dot Indicates if an endpoint is included with that particular piece of the graph Open Dot Indicates if an endpoint is not included with that particular piece of the graph Restricted Domains When the domain is restricted to include a set of specific numbers Evaluating the piecewise functions given the following domains x, if x f ( x ) x, if x x, if x. when x = 0. when x = 8. for f( ) 4. for f(0) x, if x f (x) x, if x x, if x f 0 0 f 0 f 0 0 x, if x f (x) x, if x x, if x f 8 8 f 8 f 8 f (x) x, if x x, if x x, if x f (x) x, if x x, if x x, if x f f f f 0 00 f 0 0 f 0 00
28 Graph the following functions: You are not graphing the whole line, just the part that the restrictive domain tells you to.. f x = x, if x. f x = x 5, if x 5. f x= x 7, if x 6. f x = x 5 if x 6
29 Day 0 Piecewise Functions Day Graph the following piecewise function. Objectives: SWBAT graph piecewise functions 5 x if x. g( x ) x if x x, if x. f ( x ) x, if x x. f = x 4, x x, x x, x4 4. f x= x, x 8, x 4
30 Step functions A piecewise functions that involve all horizontal lines., 7 x 5 5. f x= 0, x 4, 4 x 8 6. Write a piecewise function for the graph shown below. if 0 x f x if x if x Given the following graphs, write a piecewise function f x x if 0 x x 4 if x f x x if x 4 4 x if x
31 9. In 005, the cost C (in dollars) to send U.S. Postal Service Express Mail up to 5 pounds depended on the weight w (in ounces) according to the function below..65, if 0 x , if 8 x C w=.05, if x , if 48 x , if 64 x 80 A) Graph the function. B) What is the cost to send a parcel weighing pounds and 9 ounces? lbs 9 ozs ozs.05
Module 2 Study Guide. The second module covers the following sections of the textbook: , 4.1, 4.2, 4.5, and
Module 2 Study Guide The second module covers the following sections of the textbook: 3.3-3.7, 4.1, 4.2, 4.5, and 5.1-5.3 Sections 3.3-3.6 This is a continuation of the study of linear functions that we
More informationFormal Geometry Algebra Review
Formal Geometry Algebra Review Day Solve Linear Equations and Inequalities Objectives: SWBAT solve linear equations and inequalities. SWBAT use a formula. SWBAT isolate any variable in a formula or literal
More informationGeometry Summer Assignment 2018
Geometry Summer Assignment 2018 The following packet contains topics and definitions that you will be required to know in order to succeed in Geometry this year. You are advised to be familiar with each
More informationFOR STUDENTS WHO HAVE COMPLETED ALGEBRA 1 (Students entering Geometry)
FOR STUDENTS WHO HAVE COMPLETED ALGEBRA (Students entering Geometry) Dear Parent/Guardian and Student, Name: Date: Period: Attached you will find a review packet of skills which each student is expected
More informationAnswers to Sample Exam Problems
Math Answers to Sample Exam Problems () Find the absolute value, reciprocal, opposite of a if a = 9; a = ; Absolute value: 9 = 9; = ; Reciprocal: 9 ; ; Opposite: 9; () Commutative law; Associative law;
More informationGraphing Linear Systems
Graphing Linear Systems Goal Estimate the solution of a system of linear equations by graphing. VOCABULARY System of linear equations A system of linear equations is two or more linear equations in the
More informationChapter 7: Exponents
Chapter : Exponents Algebra Chapter Notes Name: Notes #: Sections.. Section.: Review Simplify; leave all answers in positive exponents:.) m -.) y -.) m 0.) -.) -.) - -.) (m ) 0.) 0 x y Evaluate if a =
More informationDefinition: A "system" of equations is a set or collection of equations that you deal with all together at once.
System of Equations Definition: A "system" of equations is a set or collection of equations that you deal with all together at once. There is both an x and y value that needs to be solved for Systems
More informationAlgebra I. AIR Study Guide
Algebra I AIR Study Guide Table of Contents Topic Slide Topic Slide Formulas not on formula sheet 3 Polynomials 20 What is Algebra 4 Systems of Equations 21 Math Operator Vocabulary 5 FOIL (double distribution)
More informationAnswers to the problems will be posted on the school website, go to Academics tab, then select Mathematics and select Summer Packets.
Name Geometry SUMMER PACKET This packet contains Algebra I topics that you have learned before and should be familiar with coming into Geometry. We will use these concepts on a regular basis throughout
More informationConsistent and Dependent
Graphing a System of Equations System of Equations: Consists of two equations. The solution to the system is an ordered pair that satisfies both equations. There are three methods to solving a system;
More informationSOLUTIONS FOR PROBLEMS 1-30
. Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS - 0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).
More informationSNAP Centre Workshop. Solving Systems of Equations
SNAP Centre Workshop Solving Systems of Equations 35 Introduction When presented with an equation containing one variable, finding a solution is usually done using basic algebraic manipulation. Example
More informationChapter 7: Exponents
Chapter : Exponents Algebra Chapter Notes Name: Algebra Homework: Chapter (Homework is listed by date assigned; homework is due the following class period) HW# Date In-Class Homework M / Review of Sections.-.
More informationHerndon High School Geometry Honors Summer Assignment
Welcome to Geometry! This summer packet is for all students enrolled in Geometry Honors at Herndon High School for Fall 07. The packet contains prerequisite skills that you will need to be successful in
More informationChapter 2 Linear Equations and Inequalities in One Variable
Chapter 2 Linear Equations and Inequalities in One Variable Section 2.1: Linear Equations in One Variable Section 2.3: Solving Formulas Section 2.5: Linear Inequalities in One Variable Section 2.6: Compound
More informationChapter 6. Systems of Equations and Inequalities
Chapter 6 Systems of Equations and Inequalities 6.1 Solve Linear Systems by Graphing I can graph and solve systems of linear equations. CC.9-12.A.CED.2, CC.9-12.A.CED.3, CC.9-12.A.REI.6 What is a system
More informationSection 1.4 Circles. Objective #1: Writing the Equation of a Circle in Standard Form.
1 Section 1. Circles Objective #1: Writing the Equation of a Circle in Standard Form. We begin by giving a definition of a circle: Definition: A Circle is the set of all points that are equidistant from
More informationPre-calculus is the stepping stone for Calculus. It s the final hurdle after all those years of
Chapter 1 Beginning at the Very Beginning: Pre-Pre-Calculus In This Chapter Brushing up on order of operations Solving equalities Graphing equalities and inequalities Finding distance, midpoint, and slope
More informationALGEBRA 2 Summer Review Assignments Graphing
ALGEBRA 2 Summer Review Assignments Graphing To be prepared for algebra two, and all subsequent math courses, you need to be able to accurately and efficiently find the slope of any line, be able to write
More informationAlgebra One Dictionary
Algebra One Dictionary Page 1 of 17 A Absolute Value - the distance between the number and 0 on a number line Algebraic Expression - An expression that contains numbers, operations and at least one variable.
More informationSolving Equations Quick Reference
Solving Equations Quick Reference Integer Rules Addition: If the signs are the same, add the numbers and keep the sign. If the signs are different, subtract the numbers and keep the sign of the number
More informationMA094 Part 2 - Beginning Algebra Summary
MA094 Part - Beginning Algebra Summary Page of 8/8/0 Big Picture Algebra is Solving Equations with Variables* Variable Variables Linear Equations x 0 MA090 Solution: Point 0 Linear Inequalities x < 0 page
More informationPart 2 - Beginning Algebra Summary
Part - Beginning Algebra Summary Page 1 of 4 1/1/01 1. Numbers... 1.1. Number Lines... 1.. Interval Notation.... Inequalities... 4.1. Linear with 1 Variable... 4. Linear Equations... 5.1. The Cartesian
More informationStrategic Math. General Review of Algebra I. With Answers. By: Shirly Boots
Strategic Math General Review of Algebra I With Answers By: Shirly Boots 1/6 Add/Subtract/Multiply/Divide Addmoves to the right -3-2 -1 0 1 2 3 Subtract moves to the left Ex: -2 + 8 = 6 Ex: -2 8 = - 10
More informationAlgebra 1 S1 Lesson Summaries. Lesson Goal: Mastery 70% or higher
Algebra 1 S1 Lesson Summaries For every lesson, you need to: Read through the LESSON REVIEW which is located below or on the last page of the lesson and 3-hole punch into your MATH BINDER. Read and work
More informationLesson 3-2: Solving Linear Systems Algebraically
Yesterday we took our first look at solving a linear system. We learned that a linear system is two or more linear equations taken at the same time. Their solution is the point that all the lines have
More informationThere are two main properties that we use when solving linear equations. Property #1: Additive Property of Equality
Chapter 1.1: Solving Linear and Literal Equations Linear Equations Linear equations are equations of the form ax + b = c, where a, b and c are constants, and a zero. A hint that an equation is linear is
More informationAlgebra 2 Notes Systems of Equations and Inequalities Unit 03c. System of Equations in Three Variables
System of Equations in Three Variables Big Idea A system of equations in three variables consists of multiple planes graphed on the same coordinate plane. The solutions to these systems consists of a single
More informationMath Precalculus I University of Hawai i at Mānoa Spring
Math 135 - Precalculus I University of Hawai i at Mānoa Spring - 2014 Created for Math 135, Spring 2008 by Lukasz Grabarek and Michael Joyce Send comments and corrections to lukasz@math.hawaii.edu Contents
More informationUnit 4 Systems of Equations Systems of Two Linear Equations in Two Variables
Unit 4 Systems of Equations Systems of Two Linear Equations in Two Variables Solve Systems of Linear Equations by Graphing Solve Systems of Linear Equations by the Substitution Method Solve Systems of
More informationMA.8.1 Students will apply properties of the real number system to simplify algebraic expressions and solve linear equations.
Focus Statement: Students will solve multi-step linear, quadratic, and compound equations and inequalities using the algebraic properties of the real number system. They will also graph linear and quadratic
More informationREAL WORLD SCENARIOS: PART IV {mostly for those wanting 114 or higher} 1. If 4x + y = 110 where 10 < x < 20, what is the least possible value of y?
REAL WORLD SCENARIOS: PART IV {mostly for those wanting 114 or higher} REAL WORLD SCENARIOS 1. If 4x + y = 110 where 10 < x < 0, what is the least possible value of y? WORK AND ANSWER SECTION. Evaluate
More informationMath101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2:
Math101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2: 03 17 08 3 All about lines 3.1 The Rectangular Coordinate System Know how to plot points in the rectangular coordinate system. Know the
More informationSystems of Linear Equations and Inequalities
Systems of Linear Equations and Inequalities Alex Moore February 4, 017 1 What is a system? Now that we have studied linear equations and linear inequalities, it is time to consider the question, What
More informationGeometry 21 Summer Work Packet Review and Study Guide
Geometry Summer Work Packet Review and Study Guide This study guide is designed to accompany the Geometry Summer Work Packet. Its purpose is to offer a review of the ten specific concepts covered in the
More informationNorth Carolina State University
North Carolina State University MA 141 Course Text Calculus I by Brenda Burns-Williams and Elizabeth Dempster August 7, 2014 Section1 Functions Introduction In this section, we will define the mathematical
More informationCONTENTS COLLEGE ALGEBRA: DR.YOU
1 CONTENTS CONTENTS Textbook UNIT 1 LECTURE 1-1 REVIEW A. p. LECTURE 1- RADICALS A.10 p.9 LECTURE 1- COMPLEX NUMBERS A.7 p.17 LECTURE 1-4 BASIC FACTORS A. p.4 LECTURE 1-5. SOLVING THE EQUATIONS A.6 p.
More informationA. 16 B. 16 C. 4 D What is the solution set of 4x + 8 > 16?
Algebra II Honors Summer Math Packet 2017 Name: Date: 1. Solve for x: x + 6 = 5x + 12 2. What is the value of p in the equation 8p + 2 = p 10? F. 1 G. 1 H. J.. Solve for x: 15x (x + ) = 6 11. Solve for
More informationYou try: What is the equation of the line on the graph below? What is the equation of the line on the graph below?
1 What is the equation of the line on the graph below? 2 3 1a What is the equation of the line on the graph below? y-intercept Solution: To write an equation in slope-intercept form, identify the slope
More informationEvaluate the expression if x = 2 and y = 5 6x 2y Original problem Substitute the values given into the expression and multiply
Name EVALUATING ALGEBRAIC EXPRESSIONS Objective: To evaluate an algebraic expression Example Evaluate the expression if and y = 5 6x y Original problem 6() ( 5) Substitute the values given into the expression
More informationAlgebra 2 Summer Work Packet Review and Study Guide
Algebra Summer Work Packet Review and Study Guide This study guide is designed to accompany the Algebra Summer Work Packet. Its purpose is to offer a review of the nine specific concepts covered in the
More informationAlgebra 1 Seamless Curriculum Guide
QUALITY STANDARD #1: REAL NUMBERS AND THEIR PROPERTIES 1.1 The student will understand the properties of real numbers. o Identify the subsets of real numbers o Addition- commutative, associative, identity,
More informationBooker T. Washington Summer Math Packet 2015 Completed by Thursday, August 20, 2015 Each student will need to print the packet from our website.
BTW Math Packet Advanced Math Name Booker T. Washington Summer Math Packet 2015 Completed by Thursday, August 20, 2015 Each student will need to print the packet from our website. Go to the BTW website
More informationChapter 5: Writing Linear Equations Study Guide (REG)
Chapter 5: Writing Linear Equations Study Guide (REG) 5.1: Write equations of lines given slope and y intercept or two points Write the equation of the line with the given information: Ex: Slope: 0, y
More informationMission 1 Simplify and Multiply Rational Expressions
Algebra Honors Unit 6 Rational Functions Name Quest Review Questions Mission 1 Simplify and Multiply Rational Expressions 1) Compare the two functions represented below. Determine which of the following
More informationStudy Guide for Math 095
Study Guide for Math 095 David G. Radcliffe November 7, 1994 1 The Real Number System Writing a fraction in lowest terms. 1. Find the largest number that will divide into both the numerator and the denominator.
More informationRegents Review Session #3 Functions
Regents Review Session #3 Functions A relation is a set of ordered pairs. A function is a relation in which each element of the domain corresponds t exactly one element in the range. (Each x value is paired
More informationVariables and Expressions
Variables and Expressions A variable is a letter that represents a value that can change. A constant is a value that does not change. A numerical expression contains only constants and operations. An algebraic
More informationHere are the exams I wrote when teaching Math 115 in Fall 2018 at Ferris State University. Each exam is followed by its solutions.
Here are the exams I wrote when teaching Math 5 in Fall 208 at Ferris State University. Each exam is followed by its solutions. Fall 208 Exam. (a) Find the slope of the line passing through the points
More information= = =
. D - To evaluate the expression, we can regroup the numbers and the powers of ten, multiply, and adjust the decimal and exponent to put the answer in correct scientific notation format: 5 0 0 7 = 5 0
More information1.5 F15 O Brien. 1.5: Linear Equations and Inequalities
1.5: Linear Equations and Inequalities I. Basic Terminology A. An equation is a statement that two expressions are equal. B. To solve an equation means to find all of the values of the variable that make
More informationSCIE 4101 Spring Math Review Packet #2 Notes Algebra I
SCIE 4101 Spring 011 Math Review Packet # Notes Algebra I I consider Algebra and algebraic thought to be the heart of mathematics everything else before that is arithmetic. The first characteristic of
More informationChapter 1: Precalculus Review
: Precalculus Review Math 115 17 January 2018 Overview 1 Important Notation 2 Exponents 3 Polynomials 4 Rational Functions 5 Cartesian Coordinates 6 Lines Notation Intervals: Interval Notation (a, b) (a,
More informationMath 1 Unit 1 EOC Review
Math 1 Unit 1 EOC Review Name: Solving Equations (including Literal Equations) - Get the variable to show what it equals to satisfy the equation or inequality - Steps (each step only where necessary):
More informationSUMMER REVIEW PACKET. Name:
Wylie East HIGH SCHOOL SUMMER REVIEW PACKET For students entering Regular PRECALCULUS Name: Welcome to Pre-Calculus. The following packet needs to be finished and ready to be turned the first week of the
More informationA. Incorrect! Replacing is not a method for solving systems of equations.
ACT Math and Science - Problem Drill 20: Systems of Equations No. 1 of 10 1. What methods were presented to solve systems of equations? (A) Graphing, replacing, and substitution. (B) Solving, replacing,
More informationPre-AP Algebra II Summer Packet
Summer Packet Pre-AP Algebra II Name Period Pre-AP Algebra II 2018-2019 Summer Packet The purpose of this packet is to make sure that you have the mathematical skills you will need to succeed in Pre-AP
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Order of Operations Expression Variable Coefficient
More informationPut the following equations to slope-intercept form then use 2 points to graph
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 = 16 2. 2x + y = 4 2x + y = 1 Tuesday September 23, 2014 Warm-up:
More informationBishop Kelley High School Summer Math Program Course: Algebra 2 A
06 07 Bishop Kelley High School Summer Math Program Course: Algebra A NAME: DIRECTIONS: Show all work in packet!!! The first 6 pages of this packet provide eamples as to how to work some of the problems
More informationBasic ALGEBRA 2 SUMMER PACKET
Name Basic ALGEBRA SUMMER PACKET This packet contains Algebra I topics that you have learned before and should be familiar with coming into Algebra II. We will use these concepts on a regular basis throughout
More informationFinal Exam Study Guide Dynamics of Algebra 2. Chapter Section Example
Final Exam Study Guide- 011-01 Dynamics of Algebra Chapter Section Example Chapter What is a function? Properties of a function Is it a function? Explain. Finding the Slope Slope y y m = x x 1 1 Find the
More information( ) c. m = 0, 1 2, 3 4
G Linear Functions Probably the most important concept from precalculus that is required for differential calculus is that of linear functions The formulas you need to know backwards and forwards are:
More informationPERT Practice Test #2
Class: Date: PERT Practice Test #2 Multiple Choice Identify the choice that best completes the statement or answers the question. Ê 1. What is the quotient of 6y 6 9y 4 + 12y 2 ˆ Ê 3y 2 ˆ? a. 2y 4 + 3y
More informationMath Precalculus I University of Hawai i at Mānoa Spring
Math 135 - Precalculus I University of Hawai i at Mānoa Spring - 2013 Created for Math 135, Spring 2008 by Lukasz Grabarek and Michael Joyce Send comments and corrections to lukasz@math.hawaii.edu Contents
More informationThe Circle. 1 The equation of a circle. Nikos Apostolakis. Spring 2017
The Circle Nikos Apostolakis Spring 017 1 The equation of a circle Definition 1. Given a point C in the plane and a positive number r, the circle with center C and radius r, is the locus of points that
More informationCharacteristics of Linear Functions (pp. 1 of 8)
Characteristics of Linear Functions (pp. 1 of 8) Algebra 2 Parent Function Table Linear Parent Function: x y y = Domain: Range: What patterns do you observe in the table and graph of the linear parent
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Lines and Their Equations
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 017/018 DR. ANTHONY BROWN. Lines and Their Equations.1. Slope of a Line and its y-intercept. In Euclidean geometry (where
More informationSCIE 4101 Fall Math Review Packet #2 Notes Patterns and Algebra I Topics
SCIE 4101 Fall 014 Math Review Packet # Notes Patterns and Algebra I Topics I consider Algebra and algebraic thought to be the heart of mathematics everything else before that is arithmetic. The first
More informationSummer Review. For Students Entering. Algebra 2 & Analysis
Lawrence High School Math Department Summer Review For Students Entering Algebra 2 & Analysis Fraction Rules: Operation Explanation Example Multiply Fractions Multiply both numerators and denominators
More informationACCUPLACER MATH 0311 OR MATH 0120
The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 0 OR MATH 00 http://www.academics.utep.edu/tlc MATH 0 OR MATH 00 Page Factoring Factoring Eercises 8 Factoring Answer to Eercises
More informationACCUPLACER MATH 0310
The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 00 http://www.academics.utep.edu/tlc MATH 00 Page Linear Equations Linear Equations Eercises 5 Linear Equations Answer to
More informationCP Algebra 2. Summer Packet. Name:
CP Algebra Summer Packet 018 Name: Objectives for CP Algebra Summer Packet 018 I. Number Sense and Numerical Operations (Problems: 1 to 4) Use the Order of Operations to evaluate expressions. (p. 6) Evaluate
More informationMATH 60 Course Notebook Chapter #1
MATH 60 Course Notebook Chapter #1 Integers and Real Numbers Before we start the journey into Algebra, we need to understand more about the numbers and number concepts, which form the foundation of Algebra.
More informationMath M111: Lecture Notes For Chapter 3
Section 3.1: Math M111: Lecture Notes For Chapter 3 Note: Make sure you already printed the graphing papers Plotting Points, Quadrant s signs, x-intercepts and y-intercepts Example 1: Plot the following
More informationSimplify each numerical expression. Show all work! Only use a calculator to check. 1) x ) 25 ( x 2 3) 3) 4)
NAME HONORS ALGEBRA II REVIEW PACKET To maintain a high quality program, students entering Honors Algebra II are expected to remember the basics of the mathematics taught in their Algebra I course. In
More informationWestside. Algebra 2 PreAP
Westside Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for
More informationPre-AP Algebra II Summer Packet 2014
Pre-AP Algebra II Summer Packet 014 Name: Period: PLEASE READ THE FOLLOWING!!!!!!! Wait until a few weeks before school starts to work through this packet so that the material will be fresh when you begin
More informationAlgebra 2/Trig H
Welcome to Algebra 2/Trig H 2018-2019 Welcome to Algebra 2/Trigonometry Honors! We are excited that you will be embarking on a journey to expand your understanding of mathematics and its concepts, tools,
More informationSolving 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
More informationRadicals: To simplify means that 1) no radicand has a perfect square factor and 2) there is no radical in the denominator (rationalize).
Summer Review Packet for Students Entering Prealculus Radicals: To simplify means that 1) no radicand has a perfect square factor and ) there is no radical in the denominator (rationalize). Recall the
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More informationWestside Algebra 2 PreAP
Westside Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for
More informationElementary Algebra Study Guide Some Basic Facts This section will cover the following topics
Elementary Algebra Study Guide Some Basic Facts This section will cover the following topics Notation Order of Operations Notation Math is a language of its own. It has vocabulary and punctuation (notation)
More information1: Translating Expressions
Algebra I 2017-2018 1: Translating Expressions Translate between words and math symbols, such as: The sum of twice a number and 3 Answer: 2n + 3 Also translate math symbols into verbal language. 5(x +
More informationMath 2 Variable Manipulation Part 3 Polynomials A
Math 2 Variable Manipulation Part 3 Polynomials A 1 MATH 1 REVIEW: VOCABULARY Constant: A term that does not have a variable is called a constant. Example: the number 5 is a constant because it does not
More informationBasic Fraction and Integer Operations (No calculators please!)
P1 Summer Math Review Packet For Students entering Geometry The problems in this packet are designed to help you review topics from previous mathematics courses that are important to your success in Geometry.
More informationMA 1128: Lecture 08 03/02/2018. Linear Equations from Graphs And Linear Inequalities
MA 1128: Lecture 08 03/02/2018 Linear Equations from Graphs And Linear Inequalities Linear Equations from Graphs Given a line, we would like to be able to come up with an equation for it. I ll go over
More information1 a) Remember, the negative in the front and the negative in the exponent have nothing to do w/ 1 each other. Answer: 3/ 2 3/ 4. 8x y.
AP Calculus Summer Packer Key a) Remember, the negative in the front and the negative in the eponent have nothing to do w/ each other. Answer: b) Answer: c) Answer: ( ) 4 5 = 5 or 0 /. 9 8 d) The 6,, and
More informationChapter 1: January 26 January 30
Chapter : January 26 January 30 Section.7: Inequalities As a diagnostic quiz, I want you to go through the first ten problems of the Chapter Test on page 32. These will test your knowledge of Sections.
More informationMath 1 Unit 1 EOC Review
Math 1 Unit 1 EOC Review Solving Equations (including Literal Equations) - Get the variable to show what it equals to satisfy the equation or inequality - Steps (each step only where necessary): 1. Distribute
More informationWhat students need to know for... ALGEBRA II
What students need to know for... ALGEBRA II 2017-2018 NAME This is a MANDATORY assignment that will be GRADED. It is due the first day of the course. Your teacher will determine how it will be counted
More informationCLASS NOTES: INTERMEDIATE ALGEBRA AND COORDINATE GEOMETRY
CLASS NOTES: INTERMEDIATE ALGEBRA AND COORDINATE GEOMETRY 1. The basic laws of Algebra Section 1 of these notes covers the same material as Sections 1.1, 1.2, and 1.6 of the text (in a somewhat very different
More informationSUMMER MATH PACKET for students
SUMMER MATH PACKET for students entering 9 th grade Geometry (2018) Dear Students, Welcome to Ma ayanot. We are very happy that you will be here. The teachers in the math department are looking forward
More informationPractice Calculus Test without Trig
Practice Calculus Test without Trig The problems here are similar to those on the practice test Slight changes have been made 1 What is the domain of the function f (x) = 3x 1? Express the answer in interval
More informationBasic Equations and Inequalities
Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 1 - of 45 Topic 0: Definition: Ex. 1 Basic Equations and Inequalities An equation is a statement that the values of two expressions
More informationChapter 1-2 Add and Subtract Integers
Chapter 1-2 Add and Subtract Integers Absolute Value of a number is its distance from zero on the number line. 5 = 5 and 5 = 5 Adding Numbers with the Same Sign: Add the absolute values and use the sign
More informationGaithersburg High School Summer 2018 Math Packet For Rising Algebra 2/Honors Algebra 2 Students
Gaithersburg High School Math Packet For Rising Algebra 2/Honors Algebra 2 Students 1 This packet is an optional review of the skills that will help you be successful in Algebra 2 in the fall. By completing
More informationAlgebra 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
More information