Upstream: 60 = 6(b-c) Downstream: 60 = 3(b+c) There are now two separate equations: 60 = 6b - 6c and 60 = 3b + 3c Solve both equations for b: b = 10 + c b = 10 - c Now make both equations equal each other and solve for c: 10 + c = 10 - c 2c = 0 c = 0 The speed of the current was 0 mph Now‚ plug the numbers into one of either the original equations to find the speed of the boat in still water. I chose the first equation: b = 10 + c or b = 10 + 0 b = 10 The speed of the boat in still water must
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Table for Weeks One and Two Chapter 4 Systems of Linear Equations; Matrices (Section 4-1 to 4-6) | Examples | Reference (Where is it in the text?) | | | | DEFINITION: Systems of Two Linear Equations in Two VariablesGiven the linear system ax + by = hcx + dy = kwhere a ‚ b ‚ c ‚ d ‚ h ‚ and k are real constants‚ a pair of numbers x = x0 and y = y0 [also written as an ordered pair (x0‚ y0)] is a solution of this system if each equation is satisfied by the pair. The set of all such ordered
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1.1 SOLUTIONS Notes: The key exercises are 7 (or 11 or 12)‚ 19–22‚ and 25. For brevity‚ the symbols R1‚ R2‚…‚ stand for row 1 (or equation 1)‚ row 2 (or equation 2)‚ and so on. Additional notes are at the end of the section. 1. x1 + 5 x2 = 7 −2 x1 − 7 x2 = −5 1 −2 5 −7 7 −5 x1 + 5 x2 = 7 Replace R2 by R2 + (2)R1 and obtain: 3x2 = 9 x1 + 5 x2 = 7 x2 = 3 x1 1 0 1 0 1 0 5 3 5 1 0 1 7 9 7 3 −8 3 Scale R2 by 1/3: Replace R1 by R1
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Plots Linear regression is a crucial tool in identifying and defining key elements influencing data. Essentially‚ the researcher is using past data to predict future direction. Regression allows you to dissect and further investigate how certain variables affect your potential output. Once data has been received this information can be used to help predict future results. Regression is a form of forecasting that determines the value of an element on a particular situation. Linear regression
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6 Systems Represented by Differential and Difference Equations Recommended Problems P6.1 Suppose that y 1(t) and y 2(t) both satisfy the homogeneous linear constant-coeffi cient differential equation (LCCDE) dy(t) + ay(t) = 0 dt Show that y 3 (t) = ayi(t) + 3y2 (t)‚ where a and # are any two constants‚ is also a solution to the homogeneous LCCDE. P6.2 In this problem‚ we consider the homogeneous LCCDE d 2yt + 3 dy(t) + 2y(t) = 0 dt 2 dt (P6.2-1) (a) Assume that a solution to
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1.1. Equations and Graphs In each of problems 1 - 4‚ find (a) an ordered pair that is a solution of the equation‚ (b) the intercepts of the graph‚ and (c) determine if the graph has symmetry. 1. 2. 3. 4. 5. Once a car is driven off of the dealership lot‚ it loses a significant amount of its resale value. The graph below shows the depreciated value of a BMW versus that of a Chevy after years. Which of the following statements is the best conclusion about the data? a. You should
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QUADRATIC EQUATIONS Quadratic equations Any equation of the form ax2 + bx + c=0‚ where a‚b‚c are real numbers‚ a 0 is a quadratic equation. For example‚ 2x2 -3x+1=0 is quadratic equation in variable x. SOLVING A QUADRATIC EQUATION 1.Factorisation A real number a is said to be a root of the quadratic equation ax2 + bx + c=0‚ if aa2+ba+c=0. If we can factorise ax2 + bx + c=0‚ a 0‚ into a product of linear factors‚ then the roots of the quadratic equation ax2 + bx + c=0 can be found
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Linear ------------------------------------------------- Important EXERCISE 27 SIMPLE LINEAR REGRESSION STATISTICAL TECHNIQUE IN REVIEW Linear regression provides a means to estimate or predict the value of a dependent variable based on the value of one or more independent variables. The regression equation is a mathematical expression of a causal proposition emerging from a theoretical framework. The linkage between the theoretical statement and the equation is made prior to data collection
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MC-B TOPIC; LINEAR PROGRAMMING DATE; 5 JUNE‚ 14 UNIVERSITY OF CENTRAL PUNJAB INTRODUCTION TO LINEAR PROGRAMMING Linear programming (LP; also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming
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Duality in Linear Programming 4 In the preceding chapter on sensitivity analysis‚ we saw that the shadow-price interpretation of the optimal simplex multipliers is a very useful concept. First‚ these shadow prices give us directly the marginal worth of an additional unit of any of the resources. Second‚ when an activity is ‘‘priced out’’ using these shadow prices‚ the opportunity cost of allocating resources to that activity relative to other activities is determined. Duality in linear programming
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