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Activity 3 - 1

Activity 3 - 1. Business Checking Account. Objectives. Solve a system of two linear equations numerically Solve a system of two linear equations graphically Solve a system of two linear equations using the substitution method Recognize the connections between the three methods of solution

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Activity 3 - 1

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  1. Activity 3 - 1 Business Checking Account

  2. Objectives • Solve a system of two linear equations numerically • Solve a system of two linear equations graphically • Solve a system of two linear equations using the substitution method • Recognize the connections between the three methods of solution • Interpret the solution to a system of two linear equations in terms of the problem’s content

  3. Vocabulary • System of linear equations – two equations that relate the same two variables • Numerical method – using a table of values to see with input results in the same output for the two equations • Graphical method – graph the functions and determine the coordinates of the point of intersection • Substitution method – an algebraic method using substitution to reduce the problem to one variable • Consistent– has exactly one solution (the graphs of the lines intersect) • Inconsistent – has no solutions (the graphs of the lines are parallel)

  4. Solving a System of 2 Linear Equations • Numerically – by completing a table and noting which x-value gives you the same y-value • Graphically – by graphing the equations and finding their point of intersection • Algebraically – by using properties of equality to solve the equations for one variable and then the other • Substitution method: (also known as elimination) • Addition method (the emphasis in lesson 3.3)

  5. Business Checking Account In setting up your part-time business, you have two choices for a checking account at the local bank. If you anticipate making about 50 transactions each month, which checking account will be more economical? Basic

  6. Business Checking Account • Let x represent the number of transactions. Write an equation that expresses the total monthly cost, C, for the regular account. 3. Let x represent the number of transactions. Write an equation that expresses the total monthly cost, C, for the basic account. This is a more complicated equation because the transaction fee does not apply to the first twenty checks. C = $11 + $0.17t C = $8.50 for t ≤ 20 C = $8.50 + $0.22(t – 20) = $0.22t + $4.10 for t > 20

  7. Fill in the Table • Use the two equations from the previous slideC = 11 + 0.17x and C = 4.10 + 0.22x 14.40 19.50 28.00 36.50 45.00 53.50 62.00 8.50 15.10 26.10 37.10 48.10 59.10 70.10

  8. TI-83 Table Feature • Use the table feature of your calculator to determine the x-value that produces two identical y-values • TABLE (2nd Graph) • Put in x values of interest (independent) in table • Set Y1 = to equation of interest • Go back to table to read off dependent values

  9. y x TI-83 Graph • Use your calculator to graph the functions window: Xmin: 0 Xmax: 300 Xscl: 10 Ymin = 0 Ymax = 80 Yscl = 10 Xres = 1

  10. Graphical Method • Step 1:Solve both equations for y = … • Step 2: Put into your calculator (y1 = for one and y2 = for the other) and graph (or graph by hand) • Step 3: If the lines intersect, then the intersection point is the solution; if the lines are parallel, then there is no solution; and if the lines are the same, then there are an infinite number of solutions • Step 4: Write the solution (intersection point) (use TRACE on your calculator to find it)

  11. y y y Graphical Method - Solutions • ConsistentInconsistentOne Solution No Solution  Solutions x x x

  12. Substitution Method • Step 1: Solve one or both equations for a variable (both x = … or both y = …) • Step 2: Substitute the expression that represents the variable in one equation for that variable in the other equation • Step 3: Solve the resulting equation for the remaining variable • Step 4: Substitute the value from step 3 into one of the original equations and solve for the other variable

  13. Substitution Example Given: y + x = 9 and y = 3x – 3 • Step 1: y = 9 – x and y = 3x – 3 • Step 2: 9 – x = 3x – 3 • Step 3: + x +x 9 = 4x – 3 + 3 + 3 12 = 4x 3 = x • Step 4: y + 3 = 9 or y = 3(3) – 3 y = 6 or y = 9 – 3 = 6

  14. Problem 1 Given: y = 3x – 10 and y = 5x + 14 Solve using substitution • Step 1: y = 3x – 10 and y = 5x + 14 • Step 2: 3x – 10 = 5x + 14 • Step 3: + 10 + 10 3x = 5x + 24 - 3x - 3x 0 = 2x + 24 -24 = 2x -12 = x • Step 4: y = 3(-12) – 10 = -46

  15. Problem 2 Use the substitution method to sol the following system of checking account cost functions: C = 0.17x + 11 and C = 0.22x + 4.10 • Step 1: C = 0.17x + 11 and y = 0.22x + 4.10 • Step 2: 0.17x + 11 = 0.22x + 4.10 • Step 3: -.17x -.17x 11 = 0.05x + 4.10 - 4.1 - 4.1 6.9 = 0.05x 138 = x • Step 4: 0.17(138) + 11 = 23.46 + 11 = 34.46

  16. Summary and Homework • Summary • The solution of a system of equations is the set of all ordered pairs that satisfy both equations. • The three standard methods for solving a system of equations are the Numerical method, Graphical method and Substitution method. • A linear system is consistent if there is at least one solution, the point of intersection of the graphs. • A linear system is inconsistent if there is no solution -- that is, the lines are parallel. • Homework • 1- 4, 7, 8

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