University of Phoenix Material
Learning Team Summary Worksheet TWO (due week Five) Brenda Rivera As a learning team, complete the table with formulas, rules, and examples from each section of Chapters 4, 5, 6,7,8,9,10 and 11 in the textbook. The completed summary will help prepare you for the Final Exam in Week 5. Points will be awarded for completion of the project. Study Table for Weeks One and Two
Chapter 4 Systems of Linear Equations; Matrices (Section 4-1 to 4-6)
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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 pairs is called the solution set for the system. To solve a system is to find its solution set.
| EXAMPLE 1 Solving a System by Graphing Solve the ticket problem by graphing2x + y = 8x + 3 y = 9SOLUTION An easy way to find two distinct points on the first line is to find the x and y intercepts.Substitute y = 0 to find the x intercept (2x = 8, so x = 4), and substitute x =0 to find the y intercept (y = 8).Then draw the line through (4, 0) and (0, 8).After graphing both lines in the same coordinate system (Fig. 1), estimate the coordinates of the intersection point:
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| Systems of Linear Equations: Basic TermsDefinition: A system of linear equations is consistent if it has one or more solutions andInconsistent if no solutions exist. Furthermore, a consistent system is said to be independent if it has exactly one solution (often referred to as the unique solution ) and dependent if it has more than one solution. Two systems of equations are equivalent if they have the same solution set.
| EXAMPLE 3 Solving a System Using a Graphing Calculator Solve to two decimal places using graphical approximation techniques on a graphing calculator:5x + 2y = 152x - 3y = 16SOLUTION First, solve each equation for y:5x + 2y = 152y = -5x + 15y = -2.5x + 7.52x - 3y = 16-3y = -2x + 16y =2/3x -16/3CHECKRounding the values the checks are sufficiently close but, due to rounding, not exact.5x + 2y = 15 2x - 3y = 165(4.05)+2(- 2.63)=15 2(4.05) - 3(- 2.63) =1614.99✓L 15 15.99✓L16
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| Substitution: In this method, first we choose one of two equations in a system and solve for one variable in terms of the other. (We make a choice that avoids fractions, if possible.) Then we substitute the result into the other equation and solve the resulting linear equation in one variable. Finally, we substitute this result back into the results of the first step to find the second variable.
| EXAMPLE 4 Solving a System by Substitution Solve by substitution:5x + y = 42x - 3y = 5SOLUTION Solve either equation for one variable in terms of the other; then substitute intothe remaining equation. In this problem, we avoid fractions by choosing the firstequation and solving for y in terms of x:5x + y = 4 Solve the first equation for y in terms of x.y = 4 - 5x Substitute into the second equation.2x - 3y = 5 Second equation
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| Elimination by Addition: This is probably the most important method of solution. It readily generalizes to larger systems and forms the basis for computer-based solution methods.To solve an equation such as 2x - 5 = 3, we perform operations on the equation until we reach an equivalent equation whose solution is obvious.
| EXAMPLE 5 Solving a System Using Elimination by Addition Solve the following system usingElimination by addition:3x - 2y = 82x + 5y = -1SOLUTION We use Theorem 2 to eliminate one of the variables, obtaining a system with an obvious solution:3x - 2y = 8 Multiply the top equation by 5 and the bottomequation by 2 (Theorem 2B).2x + 5y = -15(3x - 2y) = 5(8)2x + 5y = -13x - 2y = 82x + 5y = -13x - 2y...
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