...Submitted to: Mr.timothy a. acana Linear equations in two variables Forms for 2D linear equations Linear equations can be rewritten using the laws of elementary algebra into several different forms. These equations are often referred to as the "equations of the straight line." In what follows, x, y, t, and θ are variables; other letters represent constants (fixed numbers). General form where A and B are not both equal to zero. The equation is usually written so that A ≥ 0, by convention. The graph of the equation is a straight line, and every straight line can be represented by an equation in the above form. If A is nonzero, then the x-intercept, that is, the x-coordinate of the point where the graph crosses the x-axis (where, y is zero), is −C/A. If B is nonzero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis (where x is zero), is −C/B, and the slope of the line is −A/B. Standard form where A and B are not both equal to zero, A, B, and C are coprime integers, and A is nonnegative (if zero, B must be positive). The standard form can be converted to the general form, but not always to all the other forms if A or B is zero. It is worth noting that, while the term occurs frequently in school-level US algebra textbooks, most lines cannot be described by such equations. For instance, the line x + y = √2 cannot be described by a linear equation with integer coefficients since √2 is irrational. Slope–intercept form where m is...
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...Linear Functions Unit Plan Part 2 – EDCI 556 – Week 2 Darrell Dunnas Concordia University, Portland Linear Functions Unit Plan Part 2 Mr. Dunnas decides to change the graphing linear equations lesson into a problem-based lesson. This lesson is comprised of three components. Component number one is to write the equation in slope-intercept form (solve for y). Component number two is to find solutions (points) to graph via t-tables and slope-intercept form. Component number three is to graph the equation (connect the points that form a straight line). In mastering this lesson, all components must be addressed. In teaching, all learners how to graph linear equations, one must create a meaningful context for learning. First, the lesson must be aligned to the curriculum framework (Van de Walle, Karp, & Bay-Williams, 2013). Graphing linear equations is a concept found in the curriculum framework. Second, the lesson must address the needs of all students (Van de Walle, Karp, & Bay-Williams, 2013). The think-aloud strategy and graphing calculators will be used to graph linear equations and address the learning styles of all learners. Third, activities or tasks must be designed, selected, or adapted for instructional purposes (Van de Walle, Karp, & Bay-Williams, 2013). Lectures, handouts, videos, and cooperative learning activities will be used in teaching the lesson. Fourth, assessments must be designed to evaluate the lesson...
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...Equations and Inequalities Unit Overview Investigating patterns is a good foundation for studying Algebra 1. You will begin this unit by analyzing, describing, and generalizing patterns using tables, expressions, graphs, and words. You will then write and solve equations and inequalities in mathematical and real-world problems. Key Terms As you study this unit, add these and other terms to your math notebook. Include in your notes your prior knowledge of each word, as well as your experiences in using the word in different mathematical examples. If needed, ask for help in pronouncing new words and add information on pronunciation to your math notebook. It is important that you learn new terms and use them correctly in your class discussions and in your problem solutions. © 2014 College Board. All rights reserved. Academic Vocabulary • consecutive Math Terms • sequence • common difference • expression • variable • equilateral • equation • solution • formula • literal equation • compound inequality • conjunction • disjunction • absolute value • absolute value notation • absolute value equation • absolute value inequality ESSENTIAL QUESTIONS How can you represent patterns from everyday life by using tables, expressions, and graphs? How can you write and solve equations and inequalities? EMBEDDED ASSESSMENTS These assessments, following Activities 2 and 4, will give you an opportunity to demonstrate what you have learned...
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...Explain the five steps for solving rational equations. Can any of these steps be eliminated? Can the order of these steps be changed? Would you add any steps to make rational equations easier to complete or understand? - Find the LCD of the terms in the equation. - Multiply each side of the equation by the LCD. - Simplify each term. - Solve the resulting equation. - Check each answer in the given equation. Any value that makes a denominator equal 0 should be rejected because it is an extraneous solution. • You can eliminate the “check your answer” step, but it’s not recommended, because you would want to know if the answer you come up with is the correct answer. • Yes. You can simplify each term and then multiply each side by the LCD. • I will not add any steps to make the equations easier. Do all rational equations have a single solution? Why is that so? • No. Not all rational equations have a single solution. f(x) = (x² + 4x + 1)/(x^8 + 2) Then, we can get a lot of possible solutions where x can be any real value What constitutes a rational expression? How would you explain this concept to someone unfamiliar with it? - A rational expression is a fraction whose numerator and/or denominator are polynomials. - A rational expression can be written as PQ , where P and Q are polynomials. A rational expression is defined whenever Q is not equal to zero. Ex: If z = 1, then 5z + 8/z^2 - 2z + 1 = 5(1) + 8/1^2 - 2(1) + 1, or 13/0, which is undefined because division...
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... The equation relating these factors is: ∆G = ∆H–T∆S, where G is free energy, H is enthalpy, S is entropy, and T is temperature (in Kelvin). Although temperature values will always be positive, entropy, enthalpy, and free energy values can be positive or negative. For a given process, a quantitative value for each factor can be calculated using the known values of the factors for each reactant involved (see Table 1) according to the general equation ∆ X°rx = Σ X°(products)–Σ X°(reactants). See if the following activity helps you better understand what these quantities really mean. Table 1 HCO3 H+ H2O (l) CO2 (g) - ∆Η° (kJ/mol) -691.1 0 -285.8 -393.5 S° (J/K mol) 94.94 0 69.9 213.6 ∆G° (kJ/mol) -587.1 0 -237.2 -394.4 Materials • • • • • vinegar baking soda thin-walled cup tablespoon measure teaspoon measure Exploration Step 1 Put about 2 tablespoons vinegar in a cup. Add a teaspoon or two of baking soda to the cup. (a) What do you observe through sight, sound, and touch? (b) What kind of change is occurring? (c) What are the formulas of the 2 major components of vinegar and of the one component of baking soda? (d) Write the overall equation and the net ionic equation for the process. Step 2 (a) Define entropy and the significance of the sign of its value. (b) Based on your observations, explain the entropy change for the system observed in Step 1. (c) Use the entropy data from Table 1 to calculate the entropy change for the net ionic equation from Step 1. (d)...
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...2 2.1 Introduction to Equations 2.2 Linear Equations 2.3 Introduction to Problem Solving 2.4 Formulas 2.5 Linear Inequalities Linear Equations and Inequalities Education is not the filling of a pail, but the lighting of a fire. — WILLIAM BUTLER YEATS M athematics is a unique subject that is essential for describing, or modeling, events in the real world. For example, ultraviolet light from the sun is responsible for both tanning and burning exposed skin. Mathematics lets us use numbers to describe the intensity of ultraviolet light. The table shows the maximum ultraviolet intensity measured in milliwatts per square meter for various latitudes and dates. Latitude 0 10 20 30 40 50 Mar. 21 325 311 249 179 99 57 June 21 254 275 292 248 199 143 Sept. 21 325 280 256 182 127 75 Dec. 21 272 220 143 80 34 13 ISBN 1-256-49082-2 Source: J. Williams, The USA Today Weather Almanac. If a student from Chicago, located at a latitude of 42°, spends spring break in Hawaii with a latitude of 20°, the sun’s ultraviolet rays in Hawaii will be approximately 249 2.5 99 times as intense as they are in Chicago. Equations can be used to describe, or model, the intensity of the sun at various latitudes. In this chapter we will focus on linear equations and the related concept of linear inequalities. 89 Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by...
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...The first step that I did was to set up a equation with a chemical reaction like this 2Mg + O2 = 2MgO. The second step that I did was to find out which of the reactants are limiting by multiplying the number of moles of the reactant to the coefficient that is stoichiometric. The third step is to find how many moles magnesium has by forming the equation 97.2 grams * moles / 24 grams * 2 and got the answer of 8 moles The fourth step that I did was to find how many moles oxygen has by creating the equation 88.5 grams * moles / 32 grams * 1 and got the answer of 2.7 moles. Now I have discovered that oxygen has the smaller amount of moles, this is the limiting reactant. The last step that I did was to solve this problem that is based on the number of oxygen being used....
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...2 2.1 Introduction to Equations 2.2 Linear Equations 2.3 Introduction to Problem Solving 2.4 Formulas 2.5 Linear Inequalities Linear Equations and Inequalities Education is not the filling of a pail, but the lighting of a fire. — WILLIAM BUTLER YEATS M athematics is a unique subject that is essential for describing, or modeling, events in the real world. For example, ultraviolet light from the sun is responsible for both tanning and burning exposed skin. Mathematics lets us use numbers to describe the intensity of ultraviolet light. The table shows the maximum ultraviolet intensity measured in milliwatts per square meter for various latitudes and dates. Latitude 0 10 20 30 40 50 Mar. 21 325 311 249 179 99 57 June 21 254 275 292 248 199 143 Sept. 21 325 280 256 182 127 75 Dec. 21 272 220 143 80 34 13 ISBN 1-256-49082-2 Source: J. Williams, The USA Today Weather Almanac. If a student from Chicago, located at a latitude of 42°, spends spring break in Hawaii with a latitude of 20°, the sun’s ultraviolet rays in Hawaii will be approximately 249 2.5 99 times as intense as they are in Chicago. Equations can be used to describe, or model, the intensity of the sun at various latitudes. In this chapter we will focus on linear equations and the related concept of linear inequalities. 89 Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by...
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...Solution: There are many different approaches to solving this problem and I would encourage you to have a go at solving it for yourself before looking at the solutions I have presented. On the pages that follow I show two very different methods for solving this particular problem. The first method, which is a more generic method, uses loop equations and simultaneous equations to solve the problem. The second method is a very elegant solution and shows what can be achieved by using a totally different approach. It makes use of simple logical reasoning with some ohms law thrown in. Page 1 of 8 Solution 1 – Using Loop Equations The basic steps for solving this problem using loop equations are: 1. 2. 3. 4. Redraw the 3 dimensional resistive cube network in 2 dimensions. Draw all the loop currents in a clockwise direction and identify them. Write the equations for the voltage drops around each loop in turn. Solve the equations to find the unknown currents using the method of simultaneous equations. Step 1 & 2: Draw the circuit in 2d and identify the loop currents. Figure 2 Step 3: Write the voltage drop equations for each of the loops. Loop i1 (orange): V = (Ι1 +...
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...Algebra 2 Lesson 5-5 Example 1 Equation with Rational Roots Solve 2x2 – 36x + 162 = 32 by using the Square Root Property. 2x2 – 36x + 162 = 32 Original equation 2(x2 – 18x + 81) = 2(16) Factor out the GCF. x2 – 18x + 81 = 16 Divide each side by 2. (x – 9)2 = 16 Factor the perfect trinomial square. x – 9 = Square Root Property x – 9 = ±4 = 4 x = 9 ± 4 Add 9 to each side. x = 9 + 4 or x = 9 – 4 Write as two equations. x = 13 x = 5 Solve each equation. The solution set is {5, 13}. You can check this result by using factoring to solve the original equation. Example 2 Equation with Irrational Roots Solve x2 + 10x + 25 = 108 by using the Square Root Property. x2 + 10x + 25 = 108 Original equation (x + 5)2 = 108 Factor the perfect square trinomial. x + 5 = Square Root Property x = –5 ±6 Add –5 to each side; = 6 x = –5 + 6 or x = –5 – 6 Write as two equations. x ≈ 5.4 x ≈ –15.4 Use a calculator. The exact solutions of this equation are –5 – 6 and –5 + 6. The approximate solutions are –15.4 and 5.4. Check these results by finding and graphing the related quadratic function. x2 + 10x + 25 = 108 Original equation x2 + 10x – 83 = 0 Subtract 108 from each side. y = x2 + 10x – 83 Related quadratic function. CHECK Use the ZERO function of a graphing calculator. The approximate zeros of the...
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...Finding the Equation of a Tangent Line Using the First Derivative Certain problems in Calculus I call for using the first derivative to find the equation of the tangent line to a curve at a specific point. The following diagram illustrates these problems. There are certain things you must remember from College Algebra (or similar classes) when solving for the equation of a tangent line. Recall : • A Tangent Line is a line which locally touches a curve at one and only one point. • The slope-intercept formula for a line is y = mx + b, where m is the slope of the line and b is the y-intercept. • The point-slope formula for a line is y – y1 = m (x – x1). This formula uses a point on the line, denoted by (x1, y1), and the slope of the line, denoted by m, to calculate the slope-intercept formula for the line. Also, there is some information from Calculus you must use: Recall: • The first derivative is an equation for the slope of a tangent line to a curve at an indicated point. • The first derivative may be found using: A) The definition of a derivative : lim h →0 f (x + h ) − f ( x ) h B) Methods already known to you for derivation, such as: • Power Rule • Product Rule • Quotient Rule • Chain Rule (For a complete list and description of these rules see your text) With these formulas and definitions in mind you can find the equation of a tangent line. Consider the following problem: Find the equation of the line tangent to f ( x...
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...index laws are a heap of laws that are used when using calculations of indices and exponents, this helps with reducing the calculating process of potential equations that will be raised when dealing with large numbers in mathematics. Rule Example Rule Example Adding Indices when your multiply When two numbers that are the same and are been multiplied together are raised to the power a simple principle can be used to shorten the time used when achieving the problem. X^a x X^b=X^(a+b) Example:...
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...given that the final state has a temperature of 70oC. From the superheat tables for refrigerant-134a in Table A-13 on page 930, we find that the specific volume at this temperature and pressure is 0.027413 m3/kg. In order to find the volume (V in m3 as opposed to the specific volume, v, from the property tables in m3/kg), we have to know the mass. We can find the mass from the initial volume and the value of the specific volume at the initial state of saturated liquid at 900 kPa. At this pressure, we use the saturation table, A-12, on page 928, to find the specific volume of the saturated liquid, vf = 0.0008580 m3/kg at 900 kPa. We can then find the mass as follows. We can now compute V2 = mv2 and find the work done. W = 5,571 kJ 2 A mass of 2.4 kg of air at 150 kPa and 12oC is contained in a gas-tight, frictionless piston-cylinder device. The air is now compressed to a final pressure of 600 kPa. During the process, heat is transferred from the air such that the temperature inside...
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...[pic] |College of Engineering and Computer Science Mechanical Engineering Department Mechanical Engineering 370 Thermodynamics | | | |Fall 2010 Course Number: 14319 Instructor: Larry Caretto | Unit Two Homework Solutions, September 9, 2010 1 A frictionless piston-cylinder device initially contains 200 L of saturated liquid refrigerant-134a. The piston is free to move, and its mass is such that it maintains a pressure of 900 kPa on the refrigerant. The refrigerant is now heated until its temperature rises to 70oC. Calculate the work done during this process. The freely-moving piston can be interpreted as giving a constant pressure process such that P1 = P2 = P = 900 kPa. For a constant pressure process, the concept that the work is the area under the path is particularly simple. That area is a rectangle whose area is P (V2 – V1). We know that P is 900 kPa, and the initial volume is 200 L = 0.2 m3, but we have to find the final volume. Because this is a constant pressure process, the final pressure equals the initial pressure of 900 kPa (0.9 MPa) and we are given that the final state has a temperature of 70oC. From the superheat tables for refrigerant-134a in Table A-13 on page 930, we find that the specific volume at this temperature and pressure is 0.027413 m3/kg. In order to find the volume (V in m3 as opposed to the specific volume, v, from the property tables in m3/kg), we have...
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...coefficient of correlation among variables X also Y is a quantitative index of connection involving these two variables. In squared type, while a coefficient of purpose specifies the quantity of difference in the principle variable Y that is accounted for through the deviation in the analyst variable X. [pic][pic][pic][pic]Examples for Linear Regression Analysis: ABC a manufacturing co. where the production cost depends on their raw materials cost. Now, For the given set of x(tk in million) and y ( tk in thousand per unit) values, determine the Linear Regression and also find the slope and intercept and use this in a regression equation. |X |Y | |50 |4.2 | |51 |3.1 | |52 |5.1 | Solution: Step 1: let the numebr of values N = 3 Step 2: determine the values for XY, X2 |X |Y |X*Y |X*X | |50 |4.2 |210 |2500 | |51 |3.1 |158.1...
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