PreAP Algebra 2 Lesson 21 Solving 2x2 Systems. 3x 2y 22 (6,2) 5x y 28 (1, 3)


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1 Lesson 21 Solving 2x2 Systems Objectives: The students will be able to solve a 2 x 2 system of equations graphically and by substitution, as well as by elimination. Materials: paper, pencil, graphing calculator, Time 15 min Do Now: Activity 30 min Direct Instruction Background Information: Solve by substitution: 3x 4y = 10; 3x 2y = 14 (2, 4). After they figure it out, tell them that there is another way to solve linear equations that is often faster than substitution (and can avoid dealing with fractions). Concepts: Write the first equation on the board again and ask if it is ok to add various things to both sides (i.e. add 24 to both sides, add x + 7 to both sides, etc.). When students agree to this, ask them if adding 3x 2y on the left and 14 on the right is ok, and why. It is ok to do, because, since they are equal, we are adding the same thing to both sides. This method is known as elimination. The idea is to eliminate one of the variables by combining the two equations. Examples: 3x 2y 22 (6,2) 5x y 28 Concepts: 2x 3y 7 (1, 3) 4x 5y 19 7x 12y 22 (2,3) 5x 8y 14 Elimination Method 1) Multiply (if necessary) one or both of the equations by a constant to get coefficients that differ only in sign for one of the variables. 2) Add the new equations together to eliminate one of the variables and solve for the other 3) Plug in to find the other variable (or, repeat the process, eliminating the other variable) 25 min Pair Work Students practice solving systems of equations. Check with partner for correctness.
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3 Lesson 21 DO NOW *Solving 2 x 2 Systems of Equations Review In Algebra 1, you learned two ways to solve a 2 x 2 system of equations: 1. Graphically a. Convert each function into slopeintercept form (if necessary). b. Graph the two lines and find where they intersect. c. Write the coordinates of the point of intersection as an ordered pair (x, y). This is the solution to the system. 2. Algebraically (substitution method) a. Isolate y in one of the equations (i.e. get y by itself). b. Substitute what y equals into the other equation. Make sure to use the distributive property if necessary. c. Solve the new equation for x. d. Plug x back in to the original functions to get y (check by making sure you get the same y with both functions). e. Write your solution as an ordered pair (x, y). In Algebra 2, we will learn other methods to solve systems of equations. In this packet, you will just practice what you already have learned. 1) 8x 8y 16 2x 3y 9 Can you make the first equation simpler to work with? Hint: divide Solve graphically: Solve algebraically: Are they the same solutions?
4 Lesson 21 DO NOW 2) 3x y x y Can you make the second equation simpler to work with? Hint: multiply Solve graphically: Solve algebraically: Are they the same solutions? 3) To connect a DVD to a TV, you need a cable with a special connector. Suppose you can buy a 6 foot cable for $15.50, and a 3foot cable for $ Given that the cost of a cable is the sum of the cost of the connector and the cost of the cable itself, what is the price of a 4foot cable? Hint: define two variables and set up a system of equations.
5 Lesson 21 Pairwork Elimination Practice Solve each system of equations by using the elimination method. Answer the questions before you being each problem. 1. 5x y 6 5x 3y 22 a) Which variable will you eliminate? b) Which equation (1 st, 2 nd, both, or neither) will you multiply? And by what factor? 2. 3x 4y 12 6x 2y 11 a) Which variable will you eliminate? b) Which equation (1 st, 2 nd, both, or neither) will you multiply? And by what factor?
6 Lesson 21 Pairwork 3. 2x 8y 26 4x 12y 44 a) Before you start, how can you simplify these equations? Hint: divide. Then, write the simplified system below: b) Which variable will you eliminate? c) Which equation (1 st, 2 nd, both, or neither) will you multiply? And by what factor? 4. 2x 3y 12 3x 4y 1 a) Which variable will you eliminate? b) Which equation (1 st, 2 nd, both, or neither) will you multiply? And by what factor?
7 Lesson 21 Homework More Elimination Practice Solve each system of equations by using the elimination method. Work should be done on a separate sheet of paper and attached. 3x 4y 10 7x 2y x 3y 42 2x 3y x 4y 18 6x 8y x 14 4y 2x 8y x 0.05y x 0.04y x y x y y 18xy 30 13y 18xy xy y xy 3y 9. To raise money for new football uniforms, your school sells silkscreened Tshirts. Short sleeve Tshirts cost the school $8 each and are sold for $11 each. Long sleeve Tshirts cost the school $10 each and are sold for $16 each. The school spends a total of $3715 on Tshirts and sells all of them for $6160. How many of each type are sold?
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