ALGEBRA 1 UNIT 12: SYSTEMS OF LINEAR EQUATIONS

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1 Key Dates ALGEBRA 1 UNIT 12: SYSTEMS OF LINEAR EQUATIONS Quiz: Block Day, February 19-20, 2014 Test: Tuesday, February 25, 2014 Day 1 (Monday, February 17, 2014) Solving Systems of Equations through Graphing and Tables " Test I can graph a linear equation in slope intercept form. I can solve a linear equation in standard form for y to put it in slope intercept form. I can find the intersecting point of two lines and identify it as the solution of the system of equations. I can recognize that parallel lines have no solution to their system. /1 I can recognize that lines with the same slope and intercept have an infinite number of solutions to their system. I can translate a word problem into two algebraic equations. /5 Academic Practice: Grade: /7 Summary: Day 2 (Tuesday, February 18, 2014) Solving Systems of Equations Algebraically 1 - Substitution " Test I can solve linear equations for a variable of my choice. I can solve systems of linear equations using the substitution method /6 I can translate a word problem into two algebraic equations. /5 Academic Practice: Grade: /6 Summary: Day 3 (Block Day, February 19-20, 2014) Solving Systems of Equations Algebraically 2 - Elimination " Test I can convert linear equations into a form of my choosing. I can solve systems of linear equations using the substitution method /6 I can translate a word problem into two algebraic equations. /5 Academic Practice: Grade: /6 Summary: Day 4 (Friday, February 21, 2014) Solving Systems of Equations Algebraically 3 Best Method " Test I can determine the most efficient method of solving for a system of equations. I can translate a word problem into two algebraic equations. /5 Academic Practice: Grade: /5 Summary: N/A N/A /7 /1 N/A N/A N/A ALGEBRA 1 UNIT 12 WEEK 7 SYSTEMS OF LINEAR EQUATIONS NOTES & PRACTICE PAGE 1

2 Day 5 (Monday, February 24, 2014) Systems of Linear Equations Review " Test I can show mastery of the student expectations from Days 1-5. /12 Day 6 (Tuesday, February 25, 2014) Unit 12 Test: Systems of Linear Equations Grade Test I can show mastery of the student expectations from Days 1-5. /12 TABLE OR y = 1x + 5 y = x + 3 GRAPHING x y x y SUBSTITUTION Ex 1. y = x + 2 x + y = 8 Solving Systems of Equations Ex 1. 3x + 3y = 34 3x 3y = 4 ELIMINATION Ex 2. y = x + 7 4x + 2y = 4 Ex 2. 2x + 2y = 10 x + y = 4 ALGEBRA 1 UNIT 12 WEEK 7 SYSTEMS OF LINEAR EQUATIONS NOTES & PRACTICE PAGE 2

3 DAY 1: SOLVING SYSTEMS OF EQUATIONS THROUGH GRAPHING AND TABLES 1. A system of linear equations is a set of 2 or more linear equations. 2. The solution of a system is an that makes both equations. 3. On a graph of both equations, the solution is where the lines. 4. Systems of equations can have one of the following: One Solution No Solutions Infinitely Many Solutions To Solve a system of equations by graphing: 1. each line very carefully. 2. for where the lines intersect. 3. the solution in both equations. I can graph a linear equation in slope intercept form. I can find the intersecting point of two lines and identify it as the solution of the system of equations. *See pg. 384 of the textbook to see #5 & 6 worked out. 5. y = x 3 y = x 1 6. y = x y =!! x y = 2 y =!! x + 4 ALGEBRA 1 UNIT 12 WEEK 7 SYSTEMS OF LINEAR EQUATIONS NOTES & PRACTICE PAGE 3

4 I can recognize that parallel lines have no solution to their system. I can recognize that lines with the same slope and intercept have an infinite number of solutions to their system. 8. y = 3x 5 y = 3x 9. y =!! x + 4 y =!! x y = 3x + 6 y = 3x + 6 Solution: I can solve a linear equation in standard form for y to put it in slope intercept form. 11. y =!! x + 2 2x + 3y = x + 2y = 2 y = 1x 2 Create a System of Equations that would result in each of the following: 13. One Solution 14. No Solution 15. Infinitely Many Solutions ALGEBRA 1 UNIT 12 WEEK 7 SYSTEMS OF LINEAR EQUATIONS NOTES & PRACTICE PAGE 4

5 Solving Systems of Linear Equations Using a Table 1. Make a table of values 2. Find an x-value that gives the same y-value for both equations. Ex OR x y x y *The solution is: (, ) The system can have: One Solution There is one x-value that gives the same y-value for both lines. No Solution There is no x-value that gives the same y-value and slopes are the same. Infinitely Many Solutions All x-values gives the same y-values for both lines. Identify the solution and write as an ordered pair (x, y), no solution, or infinitely many solutions Solution: Solution: Solution: Solution: Identifying Solutions of Systems: Tell whether the ordered pair is a solution of the given system. *See pg. 383 of the textbook to see these worked out. 16. (4, 1); x + 2y = 6 x y = ( 1, 2); 2x + 5y = 8 3x 2y = (1, 3); 2x + y = 5 2x + y = 1 ALGEBRA 1 UNIT 12 WEEK 7 SYSTEMS OF LINEAR EQUATIONS NOTES & PRACTICE PAGE 5

6 Introduction to Word Problems with Systems: What s in My Pocket??? 1. I have nickels and dimes in my pocket that total 70 cents. How many do I have of each coin? Your guess: 2. List all possible combinations that add to $0.70 in the table below with a partner: Nickels (x) Dimes (y) Total Coins 3. Can you really know what I have in my pocket? What else would you need to know? 4. What if I told you I have a total of 11 coins? Would that determine the answer to what s in my pocket? I can translate a word problem into two algebraic equations Tables and Graphs. *See pg. 385 of the textbook to see #1 worked out. 1. Bowl-o-Rama charges $2.50 per game plus $2 for shoe rental, and Bowling Pinz charges $2 per game plus $4 for shoe rental. For how many games will the cost to bowl be the same at both places? What is that cost? Write a system of equations, one equation to represent the price at each company. Let x be the number of games played and y be the total cost. Bowl-o-Rama Bowling Pinz Total Cost is price per game times games plus shoe rental. Graph both equations on your calculator (y 1 = and y 2 =) and view the graphs of the system. Where do the two lines intersect? (Use the intersect feature of your calculator) What is the solution? Now look at the table on your calculator. Does this ordered pair appear for both equations? 2. To deliver mulch, Lawn and Garden charges $30 per cubic yard of mulch plus a $30 delivery fee. Yard Depot charges $25 per cubic yard of mulch plus a $55 delivery fee. For how many cubic yards will the cost be the same? What will that cost be? Lawn and Garden Yard Depot Total Cost is times plus What is the solution? ALGEBRA 1 UNIT 12 WEEK 7 SYSTEMS OF LINEAR EQUATIONS NOTES & PRACTICE PAGE 6

7 DAY 2: SOLVING SYSTEMS OF EQUATIONS ALGEBRAICALLY SUBSTITUTION Make a Plan: Substitution works well when: One equation is already solved for one variable in terms of another like y = or x = One equation has a term with a coefficient of 1 or -1. Make it Happen: If necessary, solve one equation for one of the variables. (Get x = or y = ) Substitute the expression for the variable in the other equation. Solve the equation. You will almost always have to distribute first. Substitute your solution into either original equation. Solve for the remaining variable. Write your answer as an ordered pair. CHECK YOUR ANSWER IN BOTH EQUATIONS! One Solution Ex: No Solutions Ex: Infinitely Many Solution Ex: I can translate a word problem into two algebraic equations Substitution. *See pg. 393 of the textbook to see #1 worked out. 1. One high-speed Internet provider has a $50 setup fee and costs $30 per month. Another provider has no set-up fee and costs $40 per month. In how many months will both providers cost the same? What will that cost be? System: Solution (ordered pair): Solution (words) ALGEBRA 1 UNIT 12 WEEK 7 SYSTEMS OF LINEAR EQUATIONS NOTES & PRACTICE PAGE 7

8 COLOR BY SOLUTION USING SUBSTITUTION Solve each system using Substitution. Find each answer and color with the given color. 1. y = 6x 11 2x 3y = 7 Solution: (, ) - Purple 2. 2x 3y = 1 y = x 1 3. y = 3x + 5 5x 4y = x 3y = 3 5x + y = 17 Solution: (, ) Light Blue Solution: (, ) - Yellow 5. y = 2 4x 3y = y = 5x 7 3x 2y = 12 Solution: (, ) Lime Solution: (, ) Teal Solution: (, ) - Blue ALGEBRA 1 UNIT 12 WEEK 7 SYSTEMS OF LINEAR EQUATIONS NOTES & PRACTICE PAGE 8

9 DAY 3: SOLVING SYSTEMS OF EQUATIONS ALGEBRAICALLY ELIMINATION Make a Plan: Elimination works well when No variable is already solved for No coefficient is 1 or -1 Make it Happen: GOAL to create additive opposites (Ex. 4x, - 4x or -7y, 7y ) If necessary, multiply one or both equations by a constant to create additive opposites. Add the two equations. This should eliminate one of the variables. Solve for the variable. Substitute into either original equations and solve for the remaining variable. Write your answer as an ordered pair. CHECK YOUR ANSWER IN BOTH EQUATIONS! One Solution Ex: No Solutions Ex: Infinitely Many Solution Ex: I can translate a word problem into two algebraic equations Elimination. *See pg. 400 of the textbook to see #1 worked out. 1. Sam spent $24.75 to buy 12 flowers for his mother. The bouquet contained roses and daisies. How many of each type of flower did Sam buy? System: Solution (ordered pair): Solution (words) ALGEBRA 1 UNIT 12 WEEK 7 SYSTEMS OF LINEAR EQUATIONS NOTES & PRACTICE PAGE 9

10 COLOR BY SOLUTION USING ELIMINATION Solve each system using Elimination. Find each answer and color with the given color. 7. 4x 2y = 12 4x + 8y = 24 Solution: (, ) - Magenta 8. 4x + 8y = 20 4x + 2y = x y = 11 2x + y = x + 5y = 1 6x + 4y = 10 Solution: (, ) Purple Solution: (, ) - Turquoise 11. 2x 9y = 25 4x 9y = x + y = 16 3x + y = 5 Solution: (, ) Orange Solution: (, ) Light Pink Solution: (, ) Lime ALGEBRA 1 UNIT 12 WEEK 7 SYSTEMS OF LINEAR EQUATIONS NOTES & PRACTICE PAGE 10

11 UNIT 12: SYSTEMS OF LINEAR EQUATIONS OVERVIEW Solutions to Systems One Solution Infinitely Many No Solution Graphically (Using a Graph) Algebraically (Using substitution or elimination) Tabular (Using a Table) Using Substitution to Solve Systems of Equations Write the Helpful Hint on pg. 390: Using Elimination to Solve Systems of Equations (p. 397, 1 st paragraph): To do this you the two equations in the system together. (p. 397, 2 nd paragraph) To get a term to eliminate you must have coefficients. More Word Problems Practice 1. Angelo runs 7 miles per week and increases his distance by 1 mile each week. Marc runs 4 miles per week and increases his distance by 2 miles each week. In how many weeks will Angelo and Marc be running the same distance? What will that distance be? System: Solution (ordered pair): Solution (words) 2. The school band sells carnations on Valentine s Day for $2 each. They buy the carnations from a florist for $0.50 each, plus a $16 delivery charge. System: Solution (ordered pair): Solution (words) ALGEBRA 1 UNIT 12 WEEK 7 SYSTEMS OF LINEAR EQUATIONS NOTES & PRACTICE PAGE 11

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