Coin Combinations Systems of Equations Students investigate what happens graphically when you add a linear combination of equations.

2-7 Solving Equations With Algebra Tiles powerpoint Practice Target Practice 4: Model with mathematics. Standard 2: Reason abstractly and quantitatively ©Evergreen Public Schools 2010

2-7 Solving Equations With Algebra Tiles powerpoint Learning Target Systems of Equations Learning Targets A-REIb I can solve a system of equations. A-REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. A-REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. How do you solve a system of equations by graphing? ©Evergreen Public Schools 2010 3

Is there a better way than More Pennies & Nickels Suppose you had a box with 519 coins (nickels and pennies) worth $17.27. How many nickels are in the box? Is there a better way than Guess & Check?

Let’s write a system of equations. Suppose you had a box with a total of 519 pennies and nickels worth $17.27. How many nickels would you have? Let’s write a system of equations. What quantities are we counting in the problem? Number of coins Value of coins

Identify the variables. Number of coins Suppose you had a box of pennies and dimes with 519 coins worth $17.27. How many nickels would you have? Identify the variables. Number of coins Let x = number of nickels Let y = number of pennies Value of coins Let 5x = value of nickels in cents Let y = value of pennies in cents Let’s write a system of equations.

There is something wrong with one of the equations. Fix it! Suppose you had a box of pennies and dimes with 519 coins worth $17.27. How many nickels would you have? Let’s solve by graphing Number of coins x + y = 519 Value of coins 5x + y = 17.27 There is something wrong with one of the equations. Fix it! 5x + y is in cents, but 17.27 is in dollars.

Let’s solve by graphing x + y = 519 5x + y = 1727 www.geogebra.org Suppose you had a box of pennies and dimes with 519 coins worth $17.27. How many nickels would you have? Let’s solve by graphing x + y = 519 5x + y = 1727 www.geogebra.org www.desmos.com Answer the question The lines intersect at (302, 217) There are 217 nickels.

x + y = 519 5x + y = 1727 Work together as a team to combine the equations and add them to the graph. Then write a conjecture. Add both equations. B. Subtract the first equation from the second equation. C. Double the first equation and add it to the second equation. D. Double the second equation and subtract it from the first equation. E. Multiply the first equation by -5 and add it to the second equation. F. Double the first equation and add it to three times the second equation. G. Make up your own combination. H. Make up your own combination. All of the combinations intersect at (302, 217)

Confirm Confirm your conjecture with the Coins In a Box problem you started with. If they don’t have one, you can give them 21 coins worth 85 cents.

penny nickel dime quarter TOTAL VALUE 1 23 6 29 53 2 19 16 35 179 3 4 9 13 229

penny nickel dime quarter TOTAL VALUE 4 16 11 27 190 5 13 27 40 805 TARGET 4 6 6 6 7 19 106 7 12 14 9 35 425