Report

Visual Algebra for Teachers Activity Set 3.7 PREP PPTX Visual Algebra for Teachers Chapter 3 REAL NUMBERS AND QUADRATIC FUNCTIONS Visual Algebra for Teachers Activity Set 3.7 Systems of Equations PURPOSE To learn: To use the substitution and elimination methods of solving systems of equations. To understand how these solutions can be viewed graphically MATERIALS Graphing calculator with table functions INTRODUCTION SYSTEMS OF EQUATIONS A system of equations is two or more equations of the same basic type. A general rule for solving a system of equations is that you need the same number of equations as “unknowns” or variables. That is, you can solve a system of two equations in two unknowns, a system of three equations in three unknowns, etc. There are two main techniques for solving a system of equations: The Substitution Method and the Elimination Method. The Substitution Method This is a system of two equations in two unknowns 2 m 3n 6 4m 4n 7 1) Solve for one variable in one equation 2 m 3n 6 2 m 6 3n m 6 3n 2 S olve for m (this is just one w ay, solving for n also w orks) The Substitution Method 2 m 3n 6 4m 4n 7 m 6 3n 2 2) Substitute the resulting value into the other equation to reduce the number of variables in the second equation. 6 3n 4 2 4m 4n 7 4n 7 m S ubstitute m 6 3n 2 into second equation The Substitution Method Simplify and then solve the new equation for the second variable (n): Substitute the value for n back into either equation 6 3n and solve for m. 4 4n 7 2 2 6 3n 4 n 7 19 2m 3 6 2 12 2 n 7 4 m 3 19 12 2 n 19 n 19 4 m 12 57 45 2 m 45 4 The Substitution Method n The final solution to this system is 19 m 2 45 4 Check your solutions: 2m 45 2 4 3n 6 19 3 6 2 2 45 6 19 4m 4n 7 45 19 4 4 7 4 2 24 90 114 24 These values create valid equations. 4 45 8 19 28 180 152 28 The Elimination Method Multiply one or more equations by a carefully chosen constant and then add the result. For example, if we multiply the first equation by -2, we can “eliminate” m when we add the equations: 1. S ystem 2 . M u ltip ly b y -2 2 m 3n 6 2 2 m 3n 2 6 4m 4n 7 4m 4n 7 3. S im plify 4 m 6 n 12 4m 4n 7 The Elimination Method 4. A dd 5 . S im p lify 4m 6n 12 n 4m 4n 7 19 2 0 2n 19 Just like with Substitution; substitute the value for n back into either equation and solve for m. n 19 2 m 45 4 Question #1 Jack and Jill and Little Boy Blue have competing lemonade stands on Rosie Lane. For c cups of lemonade and p dollars of profit, the relationship between c and p can be described using the following equations For Jack and Jill : For L ittle B oy B lue : c 10 p 20 c 5 p 25 Question #1a Write in your coursepack, Activity Set 3.7 #1a If Jack and Jill sell 30 cups of lemonade, what is their profit? (Hint: Use the J & J equation) If Little Boy Blue sells 30 cups of lemonade, what is his profit? (Hint: Use the LBB equation) Question #1b Write in your coursepack, Activity Set 3.7 #1b If Jack and Jill make a profit of $5.00, how many cups of lemonade did they sell? (Hint: Use the J & J equation) If Little Boy Blue makes a profit of $5.00, how many cups of lemonade did he sell? (Hint: Use the LBB equation) Question #1cde Write in your coursepack, Activity Set 3.7 #1cde c. For the lemonade stands, profit is a function of the cups of lemonade sold. What type of function is this? d. Solve each equation for p to determine p as a function of c. What type of function is this? Is this the same answer you had in the previous part? e. For 0 cups of lemonade, how much profit do Jack and Jill make? Little Boy Blue? What do these values, the pintercepts, mean in this context? Question #1fg Write in your coursepack, Activity Set 3.7 #1fg f. By looking at the function equations, will the functions intersect? How can you tell by just looking at the function equations? g. Graph the two functions and mark any points of intersection on the graph. Scale your axes carefully and label everything clearly. Do negative values for c make any sense here? Question #1h Write in your coursepack, Activity Set 3.7 #1h Use the Substitution Method to solve the original system of equations and determine where Jack and Jill and Little Boy Blue make the same amount of profit. Give your answers in a sentence including the correct units (cups, dollars). Question #1i Write in your coursepack, Activity Set 3.7 #1i Use the Elimination Method to solve the original system of equations and determine where Jack and Jill and Little Boy Blue make the same amount of profit. Give your answers in a sentence including the correct units (cups, dollars). Visual Algebra for Teachers You are now ready for: PREP QUIZ 3.7 See Moodle