Concept (ask yourself, “What type of **math** is involved here?” -organize the mental closet/**In** general, we need #of **variables** **equations** to solve for # of **variables** **variables**. Lesson 3 FINDING y “**IN** TERMS OF” x: If you get stuck, choosing numbers is often a great shortcut for these tough questions… Lesson 3 FINDING x USING INEQUALITIES: Just like an **equation**, but here we are solving for a range rather than just one or **two**/composed of 3 major concepts: solving **equations**, **linear** graphs, and functions. Algebra is a/

kJ, mA into A, mV into V. A quick discussion … 1.What discussions have you had with your **maths** department? 2.If you have had the discussions, what have been the key outcomes? 3.Have you discussed the / algebraically and graphically reduce a given **linear** **equation** **in** **two** **variables** to the standard form y = mx + c; calculate and interpret gradients and intercepts of graphs of such **linear** **equations** numerically, graphically and algebraicallyreduce a given **linear** **equation** **in** **two** **variables** to the standard form y = /

**In** the classical regression **equation** the response **variable** y is just a dot product of the vector representing patient characteristics ( ) and the regression weights vector ( ) which is common across all patients plus an offset b. Hyperplanes as decision surfaces A hyperplane is a **linear** decision surface that splits the space into **two**/ http://www.**math**.umn.edu/~nykamp/ **Equation** of a hyperplane A Gentle Intro to SVMs **in** Biomedicine **Equation** of a hyperplane November 2009 Consider the case of R3: An **equation** of a/

.khanacademy.org/**math**/cc-eighth-grade-**math**/cc-8th- solving-**equations**/cc-8th-**equations**-distribution/v/solving-**equations**- with-the-distributive-property https://www.khanacademy.org/**math**/cc-eighth-grade-**math**/cc-8th- solving-**equations**/cc-8th-**equations**-distribution/v/solving-**equations**- with-the-distributive-property Solving a literal **equation** uses the same steps as solving an **equation** with one **variable**. You just can not simplify all the **variables**. Steps for rewriting an **equation** **in** slope/

have high scores **in** scholastic aptitude test and vice versa. Using Cohen’s (1988) guidelines’ the effect size is large relating that there is strong relationship between **math** achievement and scholastic aptitude test. Correlation Spearman Correlation: If the assumptions for Pearson correlation are not fulfilled then consider the Spearman correlation with the assumption that the Relationship between **two** **variables** is monotonically non **linear** Example: what/

sum of the roots. The roots can be determined **in** multiple ways: quadratic formula, factoring, completing the square, etc. HS Algebra Families of Function Cycle CONTEXTS FUNCTIONS (modeling) FUNCTIONS (modeling) **EQUATIONS** (solving, manipulations **EQUATIONS** (solving, manipulations ABSTRACTION (structure, precision) ABSTRACTION (structure, precision) Families of Functions: **Linear** (one **variable**) **Linear** (**two** **variables**) Quadratic Polynomial and Rational Exponential Trigonometric Families of Functions/

**MATH** 102 Contemporary **Math** S. Rook **Linear** **Equations** **MATH** 102 Contemporary **Math** S. Rook Overview Section 7.1 **in** the textbook: Solving **linear** **equations** Graphing **linear** **equations** using intercepts Slope Slope-intercept form of a line Solving **Linear** **Equations** **Linear** **Equations** Standard Form of a **Linear** **Equation**: Ax + By = C where A, B, and C are real numbers All **variables** **in** a **linear** **equation** are raised to the first power **In** standard form when all **variables** are on one side and constants on the/

, negative, no relation) of association of data **in** scatterplots. OIM.S.1.4 Represent trends on scatterplots when appropriate, with a **linear** model. High School Mathematics A (**Math** 10) North Carolina Essential Standards Occupational Course of Study/solve systems of **linear** **equations** **in** **two** **variables**, graphically and symbolically. OMA.A.5.3 Use tables and graphs to solve pairs of **linear** inequalities **in** **two** **variables**. OMA.A.5.4 Use tables and graphs to solve systems with **linear** and exponential inequalities/

signed numbers, **linear** relationships, solving **equations**, ratio and proportion, interpreting graphs, and similarity. 7 th Grade Ed Performance Data 7 th Grade students took Ed Performance **Math** four times this year. Chapter 10: Graphing **In** this chapter, / 6: Solving **Equations** Students solved one and **two** step Algebraic **equations**. They practiced using bags and blocks to help them visualize the concept of a **variable**. Also, students used interactive whiteboard scales to keep their **equations** properly balanced./

**linear** relationships, solving **equations**, ratio and proportion, interpreting graphs, and similarity. 7 th Grade STAR **Math** Data 7 th Grade students took STAR **Math** five times this year. On average, the 7 th grade students increased their STAR **Math** Grade Equivalent by **two** and a half grade levels. The discrepancy **in**/into an **equation** for a **variable**. Students were taught to use their calculators to graph **equations** and solve systems of **two** **linear** **equations**. “I found that solving systems of **equations** was challenging /

**in** the Strength Study? Agresti/Franklin Statistics, 9 of 88 The Regression Line **Equation** When the scatterplot shows a **linear** trend, a straight line fitted through the data points describes that trend The regression line is: is the predicted value of the response **variable**/**math** and verbal SAT exams. Both exams have a mean of 500 and a standard deviation of 100. The regression **equation** relating y = **math**/ of the residuals is not normal However…. **Two**-sided inferences about the slope parameter still work/

get the same result using both approaches!! **Linear** regression has the advantages of: Allowing multiple predictors (tomorrow) Accommodating continuous predictors (relationship to correlation) Accommodating categorical predictors (tomorrow) Very flexible approach Conclusion Indicator **variables** can be used to represent dichotomous **variables** **in** a regression **equation** Interpretation of the coefficient for an indicator **variable** is the same as for a continuous **variable** Provides a group comparison Tomorrow we will/

**in** Bremen, Germany Died: 1927 **in** Göttingen, Germany BMayer@ChabotCollege.edu ENGR-25_Lec-23_ODEs_Euler_Numerical.pptx 35 Bruce Mayer, PE Engineering/**Math**/Physics 25: Computational Methods Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Engr/**Math**/Physics 25 Appendix BMayer@ChabotCollege.edu ENGR-25_Lec-23_ODEs_Euler_Numerical.pptx 36 Bruce Mayer, PE Engineering/**Math**/Physics 25: Computational Methods 2 nd Order **Linear** **Equation**/ the **Two** INITIAL Conditions to generate **two** Eqns /

Geodesics (Pure **Math**) Differential **Equations** **Equations** that describe rates of change **Equations** that involve derivatives: (Falling object) (Logistic **Equation**) (Wave **Equation**) (Current) **Two** Types Ordinary (Only One Independent **Variable**) Partial (Multiple Independent **Variables**) (Only One Independent **Variable**) This Class Ordinary (Only One Independent **Variable**) Order of a Differential **Equation**: Order of Highest Derivative Second Order First Order Second Order Fifth Order Second Order (PDE) **Linear** and Nonlinear/

**in** right triangles **in** real- world and mathematical problems **in** **two** and three dimensions. 8.G.8 Apply the Pythagorean Theorem to find the distance between **two** points **in** a coordinate system. A-CED.1 Create **equations** and inequalities **in** one **variable** and use them to solve problems. Include **equations** arising from **linear**/, 3-4! http://www.khanacademy.org/**math**/algebra/linear_inequa lities/inequalities/v/multi-step-inequalities http://www.khanacademy.org/**math**/algebra/linear_inequa lities/inequalities/v/multi-step-/

, is on the left of the line. The point A with coordinates (-4, 3) is **in** Region 1. Theory – Inequalities **in** **Two** Dimensions 30S Applied **Math** Mr. Knight – Killarney School Slide 3 Unit: **Linear** Programming Lesson: Graphing Inequalities **in** **Two** **Variables** If the x-coordinate of point A, -4, is substituted into the **equation** of the line to find the corresponding y-coordinate on the line, the result is/

coefficient matrix: augmented matrix: I. Systems of **Linear** **Equations** --------------------------------------------------------------------------------------------------------------------------- I. System of **Linear** **Equations** The **variable** corresponding to pivot columns **in** the matrix are called basic **variable**. The other **variable** is called free **variable**. Definition of Basic and Free **variable**. row operations x 1, x 3, x 4 : basic **variables** x 2 : free **variables** If, then the system AX = b is/

MTH108 Business **Math** I Lecture 4 Chapter 2 **Linear** **Equations** Objectives Provide a thorough understanding of the algebraic and graphical characteristics of **linear** **equations** Provide the tools which allow one to determine the **equation** which represents a **linear** relationship Illustrate some applications Today’s Topics Importance of **linear** mathematics Characteristics of **linear** **equations** Solution set **Linear** **equations** with n **variables**; solution set and examples Graphing **linear** **equation** of **two** **variables** Solution set,/

Basic **Math** Review 1b. Simplifying Algebraic Expressions 1c. Solving **Linear** **Equations** **in** One **Variable** 1d. Solving Absolute Value **Equations** 1e. Applications of **Linear** **Equations** Unit 2: **Linear** Inequalities; Graphing **Linear** **Equations** **in** **Two** **Variables** Overview - Teaching Insights & Strategies 2a. Solving **Linear** Inequalities 2b. Cartesian Coordinate System 2c. Graphing **Linear** **Equations** **in** **Two** **Variables** 2d. Graphing: Slope-Intercept Form 2e. Graphing: Point-Slope Form 2f. Applications of **Two** **Variable** **Equations** 26/

by : Type,Order,**Linearity**, Classifiation by Type: Ordinary Differential **Equation** If a Differential **Equations** contains only ordinary derivatives of one or more dependent **variables** with respect to a single independent **variables**, it is said to be an Ordinary Differential **Equation** or (ODE) for short. For Example, Partial Differential **Equation** If a Differential **Equations** contains partial derivatives of one or more dependent **variables** of **two** or more independent **variables**, it is said/

) to degrees Fahrenheit (y or F). So, when F = 94°F, we sub this into the **equation** and calculate that C = 34.4°C The temperature **in** Dallas, Texas is 94°F. What is that temperature **in** degrees Celsius? (34.4, 94) Cell Phone Bill – A **linear** function What are **two** **variables** involved **in** simple cell phone billing system? monthly usage (minutes) – x because this is the/

for the day. 8 th grade 2.2 Summarize BIG IDEAS What **two** values are important when writing a **linear** **equation**? Which of these **two** is easier to find, **in** your opinion? **In** an **equation** of the form, y= mx + b, what does each letter /**variable**? How can you use an **equation** to find the value of the other **variable**? * 8 th grade Launch 2.4 “Intersecting **Linear** Models” Goal: Use **equations** to represent questions about problem situations and interpret the solutions **in** the context of the problem Do NOT take out your **math**/

one **equation** for one of the **variables** – Substitute (replace) into the OTHER **equation** DO NOT substitute into the **equation** that was just used to solve for a **variable**! – What results is an **equation** **in** 1 **variable** – After solving this **equation**, we now have a partial coordinate – Evaluate the partial coordinate into either **equation** to get the full coordinate Often easier to plug into the substitution Solving **Linear** Systems of **Equations** by Substitution/

**in** the Strength Study? Agresti/Franklin Statistics, 9 of 88 The Regression Line **Equation** When the scatterplot shows a **linear** trend, a straight line fitted through the data points describes that trend The regression line is: is the predicted value of the response **variable**/**math** and verbal SAT exams. Both exams have a mean of 500 and a standard deviation of 100. The regression **equation** relating y = **math**/ 88 Testing Independence between Quantitative **Variables** Steps of **Two**-Sided Significance Test about a /

**Linear** Inequalities **in** **Two** **Variables** 2.5 2.6 - Transforming **Linear** Functions 2.6 2.7 - Curve Fitting with **Linear** Models 2.7 2.8 – Solving Absolute Value **Equations** and Inequalities 2.8 2.9 - Absolute Value Function 2.9 2.5 - **Linear** Inequalities **in** **Two** **Variables** Algebra II **Linear** functions form the basis of **linear** inequalities. A **linear** inequality **in** **two** **variables** relates **two** **variables**/than 3 units from zero. The solution is a disjunction: x 3. 2.8 **Math** Joke Q: Why are you so negative? A: Just take me for my /

, weak, moderate? MEASURING **LINEAR** ASSOCIATION Scatterplots (bi-variate data) show direction, outliers/ deviation(s), form, strength of relationship between **two** quantitative **variables** **Linear** relationships are important; common, simple pattern; **linear** relationships are our focus **in** this course **Linear** relationship is strong if points/**equation** of the LSRL? # Hours Spent Studying for the Stats Test Percentage Earned on the Stats Test 389 492 4.594 185 1.586 183 1.592 CLASS ACTIVITY… 1.Groups of 2; go to COC **Math**/

**Maths** year 1, A-level **Maths** year 2 (2017 specification) Revision Binomial Theorem, Sequences & Series Functions & Transformations Trigonometry & Circular Measure Further Differentiation Further Integration Numerical Methods Partial Fractions & Integration Parametric **Equations** Trigonometry Differential **Equations** Kinematics **in** **Two**/notes: B4 [Solve simultaneous **equations** **in** **two** **variables** by elimination and by substitution, including one **linear** and one quadratic **equation**] B6 [Manipulate polynomials /

Quadratic, square root, inverse variation Absolute value, polynomial, rational, sine/cosine Solving **equations**/inequalities **in** one **variable** 6 th grade – understand a solution as the value(s) that make a **math** sentence true, solve one-step **linear** **equations**, interpret x c 7 th grade – solve **two**-step **linear** **equations** and inequalities **in** one **variable** 8 th grade – solve **linear** **equations** with rational coefficients, solve x 2 = p and x 3 = p with roots/

Chapter 3 **Linear** Systems **In** Chapter 3, You Will… Learn to solve systems of **equations** and inequalities **in** **two** **variables** algebraically and by graphing. Learn to graph points and **equations** **in** three dimensions. Learn to solve systems of **equations** **in** three **variables**. 3-1 Graphing Systems of **Equations** What you’ll learn … To solve a system by graphing 2.10 Use systems of **two** or more **equations** or inequalities to model and solve problems; justify/

as an ordered pair of numbers Substitution Method: Example Solving Systems of **Linear** **Equations** algebraically Substitution Method: Example To solve a system of **linear** **equations** using substitution: Isolate one of the **variables** on one side of one of the **two** **equations**. 2x + 5y = 7 x + 4y = 2 x = 2 - 4y Go to the other **equation** and **in** place of the **variable** isolated **in** step 1, substitute the expression equal to that/

problems that involve systems of **linear** **equations** **in** **two** **variables**, graphically and algebraically. Polynomials Multiply polynomials Factorize various polynomials using different methods Roots and Powers Simplify algebraic expressions with rational exponents Relations and Functions Identify and represent **linear** relations **in** different ways Use intercepts, rate of change, domain and range to describe graph **Linear** Functions Graph of **linear** function to its **equation** **in** slope-intercept form. Graph of/

Transition to Common Core Grade Level Expectations and Content **Math** Placement Process Options for classes/levels and High School Grade/**two** categorical and quantitative **variables**. Interpret **linear** models. Expressions and **Equations** Interpret the structure of expressions. Write expressions **in** equivalent forms to solve problems. Perform arithmetic operations on polynomials. Create **equations** that describe numbers or relationships. Solve **equations** and inequalities **in** one **variable**. Solve systems of **equations**/

solve one- and **two**-step **linear** **equations** 8.15 a) solve multistep **linear** **equations** 8.16 graph **linear** **equations** **in** **two** **variables** Alg1.4 solve multistep **linear** and quadratic **equations** (2 **variables**) Alg1.5 solve multistep **linear** inequalities (2 **variables**) Alg1.6 graph **linear** **equations** and inequalities 5/ Heinemann Mathematics (not free, but cheap) http://books.heinemann.com/**math**/ http://books.heinemann.com/**math**/ 21 st Century Skills **Math** ITRTs Michaelanne Day, Highland Springs HS Jennifer Maddux, Short Pump MS/

and a proportion is an **equation** stating that **two** ratios are equal. **In** a proportion, the cross products are equal. If a proportion contains a **variable**, you can cross multiply to solve for the **variable**. When you set the cross products equal, you create a **linear** **equation** that you can solve by using the skills that you learned **in** Lesson 2-1. Reading **Math** **In** a ÷ b = c ÷ d, b/

GRAPHING **LINEAR** **EQUATIONS** MSJC ~ San Jacinto Campus **Math** Center Workshop Series Janice Levasseur GRAPHING **LINEAR** **EQUATIONS** MSJC ~ San Jacinto Campus **Math** Center Workshop Series Janice Levasseur Basic Definitions Axes – perpendicular number lines x-axis – horizontal number / A and B are not both 0 Graphing a **Linear** **Equation** (Plotting Points) 1.Solve the **equation** for one of the **variables**, usually y 2.Pick a value for x, plug it **in**, & solve for y 3.Repeat at least **two** more times 4.Plot the points on the same /

**linear** **equations** to real world problems. Dependent **Variable** Learning how to work **linear** **equations** **in** traditional lecture- based classroom Assessment Questionnaire All students were given a Likert- style questionnaire. The questionnaire queried student perception of success for **math** **in** general and specific **math**/iMovies Data Sampling method would be Comprehensive using both of our **two** classes. Sampling method would be Comprehensive using both of our **two** classes. Data collected would be a survey on what algebra/

to -) Reflection => **two** times the projection distance /**in** OFDM. Quadratic forms: xTAx **Linear**: y = mx + c … generalizes to vector **equation** y = Mx + c (… y, x, c are vectors, M = matrix) Quadratic expressions **in** 1 **variable**: x2 Vector expression: xTx (… projection!) Quadratic forms generalize this, by allowing a **linear**/**in** communications **math** (MIMO). They will also appear **in** SVD (singular value decomposition) The pseudo inverse (ATA)-1 AT will appear **in** decorrelator receivers for MIMO More: http://tutorial.**math**/

Wronskian **Math** for CS Properties of the Wronskian (1) If y1 and y2 are **two** solutions of the **equation** y + p(x)y + q(x)y = 0, then (2) If y1 and y2 are **two** solutions of the **equation** y + p(x)y + q(x)y = 0, then **In** this case, we say that y1 and y2 are **linearly** independent. (3) If y1 and y2 are **two** **linearly** independent/

exit ticket 7. Algebraic **Equation** What does this word mean to you? This word actually means….. a **math** sentence that shows **two** equivalent expressions using **variables** and numbers. Examples… Explain how you would use this word **in** **math**. Lesson SPI 0706.2./Examples… Explain how you would use this word **in** **math**. Lesson SPI 0706.3.6 Solve **linear** **equations** with rational coefficients symbolically or graphically. SPI 0706.3.8 Solve contextual problems involving **two**-step **linear** **equations**. N2K 2 Need to Know 2 Friday: /

from both sides so is on one side. Step 3: Factor and solve the **two** **linear** **equations**. or HAWKES LEARNING SYSTEMS **math** courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Example: Solving Quadratic **Equations** by Factoring Solve the quadratic **equation** by factoring. or **In** this example, the **two** **linear** factors are the same. **In** such cases, the single solution is called a double solution or a double root/

incorporate the costs due to shortages **Two** possible modeling approaches: The first is the more “natural”, but it results **in** a nonlinear model The second is more complicated, but allows for a **linear** model Backlogging – Nonlinear Approach /**equation** **in** the Solver dialog box. Model Formulation Facility Location and Logistics Planning at Huntco Upper limit on amount shipped out of each warehouse: For each warehouse we need a constraint of the form Total shipped out of warehouse UpperBound * Fixed-cost **variable**/

line or part of a nonvertical line. Function f(x): A relationship that pairs one input to exactly one output (x, y) GOAL: **In** **math** we use tables, words, **equations**, set of ordered pairs and graphs to represent a relationship between **two** **variables**. 1. Geometric Relationships, 2. **Linear** Functions This can be done when we are presented with the following: GEOMETRIC RELATIONSHIPS: Ex: Use words, an/

the -intercept of the line is. If the **variable** does not appear **in** the **equation**, the slope is 0 and the **equation** is simply of the form. HAWKES LEARNING SYSTEMS **math** courseware specialists Copyright © 2011 Hawkes Learning Systems. All /Intercept Form Find the **equation**, **in** slope-intercept form, of the line that passes through the **two** points and. HAWKES LEARNING SYSTEMS **math** courseware specialists Copyright © 2011 Hawkes Learning Systems. All rights reserved. **Linear** Regression o **In** many practical applications, /

Research 2004 **Math**-related Program Clusters/ **Equations** /**in** Acute Triangles Measurement and Trigonometry –Optimization Problems –Solving Problems Involving Trigonometry Reasoning With Data –Working with One **Variable** Data -Applying Probability Reasoning With Data –**Two** **Variable** Analysis –Evaluating Validity 2000 CURRICULUM Grade 11M: Functions 2006 DRAFT REVISED CURRICULUM Grade 11U: Functions and Applications define the term function; explain the meaning of the term function, through investigation of **linear**/

**Linear** **Equations** **Two** or more lines. Can solve graphically, algebraically, or by substitution. Three possibilities: 1. The **two** lines intersect. Point of intersection is the solution. 2. No solution; the lines are parallel. 3. Infinite solutions; they are the same line. Substitution Method Write one **equation** so that it equals the **variable**. Substitute that **equation** for the **variable** **in** the second **equation**. Solve for the **variable** Insert solution for **variable** into either **equation**/

+ B = B + C A. B = B. A **LINEAR** **EQUATIONS** WITH ONE **VARIABLE** PRIOR KNOWLEDGE for one **variable** **LINEAR** **equation** lesson 1.What is a **equation**?. ax + b = 0 **Two** expressions set equal to each other **VARIABLE** WHAT IS A **VARIABLE** X,Y,5X,4Y,8A SOLVE **LINEAR** **EQUATIONS** WITH ONE **VARIABLE** **Linear** **Equation** Ex: X + 5 = 10 Step 1: X + 5 = 10 STEP 1 :- Identify the **variable** **in** the **equation**. **Variable** CCSS.**MATH**.CONTENT.8.E.E.7.A X/

and graphing **linear** **equations** Solving and graphing **linear** **equations** The language of inequalities The language of inequalities Our Proposal – Technology Use My **Math** Lab, Hawkes Learning, or similar programs with both courses Use My **Math** Lab, Hawkes/ 9.1 Compound Inequalities 9.1 Compound Inequalities 9.4 **Linear** Inequalities **in** **Two** **Variables** and Systems of **Linear** Inequalities 9.4 **Linear** Inequalities **in** **Two** **Variables** and Systems of **Linear** Inequalities 10. Radicals, Rational Exponents 10. Radicals, Rational /

single-step **linear** **equations** using inverse operations. 0505.3.1 Evaluate an expression by substituting a number for the **variable** or **variables** **in** the problem. Objective: Understand and be able to solve expressions and **equations**. What is a **variable**??? Discuss with a neighbor These are **variables** These are/scored 117 more points than Seth when they went bowling. The **math** expression n + 117 represents this sentence. What does "n" stand for **in** this **math** expression? A. Alex and Seths score together B. Seths score /

2x – 8 - 8x -6x +3 = -6x -8 3 = -8 2.3 Several **linear** **equations** **in** one **variable** Systems of **Equations** Solving systems of **equations** with **two** or more **linear** **equations** Substitution Elimination Cramer’s Rule Graphical Representation The 3 possible solutions still occur. 1.The solution is a /**math** problems written **in** word form General Rule: Like reading English - Left to Right Special Cases: Change **in** order terms some time called “turnaround” words (Cliff Notes: **Math** Word Problems, 2004) Convert into **Math**… **Two**/

student solves: c. Systems of **linear** **equations** with **two** unknowns using integer coefficients and constants. 6. Solve systems of **linear** **equations** exactly and approximately (e.g., with graphs), focusing on pairs of **linear** **equations** **in** **two** **variables**. 2.2.K.3 – The student solves: c. Systems of **linear** **equations** with **two** unknowns using integer coefficients and constants. 7. Solve a simple system consisting of a **linear** **equation** and a quadratic **equation** **in** **two** **variables** algebraically and graphically. For example/

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