Today s Agenda E — How can graphs

Today’s Agenda • E - How can graphs be used to solve systems of linear equations? • A - Warm-up: Bell Ringer Patterns B • Review: Graph linear equations from slope-intercept form (graphic organizer) • T - New: System of Linear Equations (graphic organizer and video) – Class work: A Good Deal Shoulder Buddy Practice Problem • S – 3 -2 -1 Homework: Graphing. Smyrna High School Equations worksheet #1 -6 3/16/2018 Geometry CP 1

Warm-up Bell Ringer Day 1~Discovering Patterns… • Complete the pattern with your group. Then answer the questions as best you can. 3/16/2018 Smyrna High School Geometry CP 2

Pattern B • Draw the penny pattern in figure 4. • How many pennies are in figure 5? • How many pennies are in figure 10? • What figure has 63 pennies? • 3/16/2018 Write a rule for this pattern? Smyrna High School Geometry CP 3

Unit 1 – Lesson 3: Systems of Linear Equations Systems of Equations Two or more linear equations that have to be solved simultaneously. 3/16/2018 Smyrna High School Geometry CP 4

Check Skills You Will Need • • How to Graph an Equation in Slope-Intercept Form y = mx + b In the slope-intercept form of a linear equation, each letter represents a special value. m = the slope of the equation (steepness). b = the y-intercept (starting number). With these two facts, an equation in slope 3/16/2018 Smyrna High School intercept form can be graphed. Geometry CP 5

Check Skills You Will Need How to Graph an Equation in Slope-Intercept Form y = mx + b • With these two facts, lets look at some examples. • y = 4 x + 2 • First locate point b on the graph and mark it. Since this is the point where the graph of the equation crosses the y-axis, the x value will be zero and the y value will be b in the equation 3/16/2018 Smyrna High School 6 Geometry CP (0, b).

Check Skills You Will Need How to Graph an Equation in Slope-Intercept Form y = mx + b • Second, determine the slope (m) of the equation. • What number is before the x in the equation? • If there is not a number in front of the letter x, remember the number 1 is understood. 3/16/2018 Smyrna High School Geometry CP 7

Check Skills You Will Need How to Graph an Equation in Slope-Intercept Form y = mx + b • Slope is commonly know as the rise/run. • The slope of the example can be written as 4/1. • Use these values to plot some more points. 3/16/2018 Smyrna High School Geometry CP 8

Check Skills You Will Need • • How to Graph an Equation in Slope-Intercept Form y = mx + b Start at the y-intercept (0, 2). From this point use the slope of the equation to find a second point. The slope of the line is 4/1 so count up 4 spaces (the rise) and count to the right 1 space (the run). Repeat these steps to. Smyrna High School find a third point. 3/16/2018 9 Geometry CP

Check Skills You Will Need How to Graph an Equation in Slope-Intercept Form y = mx + b • Remember, the slope of the line will determine the direction of the rise and run. • Going up is a positive direction. Then run to the right. • Going down is a negative direction. Then run to the right. 3/16/2018 Smyrna High School Geometry CP 10

Check Skills You Will Need How to Graph an Equation in Slope-Intercept Form y = mx + b • Remember, a negative divided by a negative is a positive. • Positive slopes will start lower on the left side of the graph and end up higher on the right side of the graph. • Negative slopes will start higher on the left side of the graph and end up. Smyrna High School lower on the right side of the 3/16/2018 11 Geometry CP graph.

Pattern C • How many toothpicks are in figure 5? • How many toothpicks are in figure 10? • What figure has 44 pennies? • Write a rule for this pattern. 3/16/2018 Smyrna High School Geometry CP 12

Check Skills You Will Need How to Graph an Equation in Slope-Intercept Form y = mx + b • For practice, graph the following equations in slopeintercept form. 1. y = -x + 6 2. y = 2 x – 4 3. y = ¼ x 3/16/2018 Smyrna High School Geometry CP 13

(0, 6) -1/1 (0, - 4) 2/1 (0, 0) 3/16/2018 1/4 Smyrna High School Geometry CP 14

Graph! • http: //www. youtube. com/watch? v=2 BHz. XIt k. By. U 3/16/2018 Smyrna High School Geometry CP 15

Ticket In The Door Recap • For your equation do the following: a) Circle & Identify the y-intercept (don’t forget to circle the sign also) b) Graph the y-intercept (always on the y-axis) c) Identify the slope (write it as a fraction) d) Use slope to plot more points (ryse/run) e) Draw a line through the points You may use your “yellow” graphic organizer to help guide you through the steps! 3/16/2018 Smyrna High School 16 Geometry CP

3/16/2018 Smyrna High School Geometry CP 17

Homework check • Use the red pen to make corrections to your homework • Now, is NOT the time to do your homework • I should not see any writing utensils except the red pen • If you didn’t do your homework then you can turn it in tomorrow for half credit kuta graphing lines in slope intercept form hw. pdf 3/16/2018 Smyrna High School Geometry CP 18

Systems of Equations • Let’s take a look at a few examples. • Your task: I Notice, I Wonder – Discuss and write down some things that you NOTICE about the examples. NOTICE – Discuss and write down some things that you WONDER about the WONDER examples. 3/16/2018 Smyrna High School Geometry CP 19

Smyr na High Scho ol Geom etry CP 3/16/2018 20 Systems of Equations: Example 1

Systems of Equations: Example 2 3/16/2018 Smyrna High School Geometry CP 21

Systems of Equations: Example 3 3/16/2018 Smyrna High School Geometry CP 22

3 Methods to Solve System of Equations • Graphing Method • Substitution Method • Elimination Method 3/16/2018 Smyrna High School Geometry CP 23

3 Possible Solution Types • 1 Exact Solution • No Solution • Infinitely Many Solutions 3/16/2018 Smyrna High School Geometry CP 24

1 Exact Solution Point 3/16/2018 of int Smyrna High School Geometry CP ersec tion 25

Another type of solution • How would you describe these lines? Y = 3 x + 2 Y = 3 x - 4 3/16/2018 What do you think the solution Smyrna High School 26 (point of intersection) is? Geometry CP

No Solution Parallel Lines Will not intersect 3/16/2018 same slope different yintercepts Smyrna High School Geometry CP 27

Another type of solution • What do you notice about the graphs and equations? Y = -3 x + 4 3 x + y = 4 What do you think the solution (point of intersection) is? 3/16/2018 Smyrna High School Geometry CP 28

Infinitely Many Solutions SAME LINE They are the SAME equation (just written in different forms) They have same slope and same y-intercept. 3/16/2018 Smyrna High School Geometry CP 29

The Goal of Solving Systems • To find one pair (x, y) of values that satisfies both linear equations. – The one pair of values that makes both equations true. 3/16/2018 Smyrna High School Geometry CP 30

Hamilton High School 3/16/2018 Smyrna High School Geometry CP 31

Hamilton High School • Students in the Hamilton High School science club want to raise $240 to buy a special eyepiece for the highpowered telescope at their school. The PTA offered to pay club members for an after-school work project that would clean up a nearby park and recreation center building. • Because the outdoor work was harder and dirtier, the deal with the PTA would pay $20 for each outdoor worker ad $10 for each indoor worker. The club had 18 members eager to work on the project. 3/16/2018 Smyrna High School Geometry CP 32

Hamilton High School • What do the variables x and y represent in these equations? • What problem condition is represented by the equation 20 x + 10 y = 240? • What problem condition is represented by the equation x + y = 18? • Determine 3 combinations of numbers of outdoor and indoor workers that will allow the club to earn just enough money to buy the telescope eyepiece? Smyrna High School 3/16/2018 33 Geometry CP

Think-Pair-Share Let’s Look at the Solution Complete Problem 1 - Parts a, b, c & d. Try it on your own Think: 5 minutes Check with a partner Pair: 2 minutes Whole group discussion Share: 5 minutes 3/16/2018 Smyrna High School Geometry CP 34

Hamilton High School • What does the x represent? x: # of outdoor workers • What does the y represent? • y: # of indoor workers 3/16/2018 Smyrna High School Geometry CP 35

Hamilton High School 20 x + 10 y = 240 3/16/2018 • What does this equation represent in the problem? • 20 x + 10 y = 240 shows the amount of money that can be earned depending on the # of outdoor and indoor Smyrna High School 36 workers Geometry CP

Hamilton High School x + y = 18 • What does this equation represent in the problem? • x + y = 18 shows that the # of club members who will work 3/16/2018 Smyrna High School Geometry CP 37

Let’s finish the problem • Now, take a look at your 3 combination of workers from part d. • Will any of the combinations also put each willing club member to work? 3/16/2018 Smyrna High School Geometry CP 38

3/16/2018 Smyrna High School Geometry CP 39

Today’s Agenda • E - How can graphs be used to solve systems of linear equations? • A - Warm-up: Bell Ringer Patterns • Review: Graph linear equations from slope-intercept form (graphic organizer) • T - New: Graph linear equations from intercepts (graphic organizer) – Class work: A Good Deal Shoulder Buddy Practice Problem, Verifying solutions, Solving systems • S – Journal entry 3/16/2018 Smyrna High School Geometry CP Homework: Pattern D 40

WARMUP: Pattern E • • • Draw the pattern for figure 5 How many white tiles are in figure 5. How many white tiles are in figure 8? What figure has 72 white tiles. Write a rule for the pattern. 3/16/2018 Smyrna High School Geometry CP 41

HOMEWORK: Pattern D • How many toothpicks are in figure 5? • How many toothpicks are in figure 10? • What figure has 44 pennies? • Write a rule for this pattern. 3/16/2018 Smyrna High School Geometry CP 42

How Do We Graph a Linear Equation? ? ? One way to graph a linear equation is in the form y = mx + b, where m is the slope and b is the y-intercept 3/16/2018 Smyrna High School Geometry CP 43

How Do We Graph a Linear Equation not in y=mx+b form? ? ? Let’s Practice: -3 x + 2 y = 12 • Since it is not in y = mx + b, we will find the x- and y-intercepts and graph those points. • Find the x-intercept: Let y = 0 and solve the equation for x. • Find the y-intercept: Let x = 0 and solve the equation for y. 3/16/2018 Smyrna High School Geometry CP 44

How Do We Graph a Linear Equation? ? ? Let’s Practice: -3 x + 2 y = 12 • Find the x-intercept: Let y = 0 and solve the equation for x. -3 x + 2(0) = 12 -3 x = 12 -3 -3 x = -4 or (-4, 0) 3/16/2018 Smyrna High School Geometry CP 45

How Do We Graph a Linear Equation? ? ? Let’s Practice: -3 x + 2 y = 12 • Find the y-intercept: Let x = 0 and solve the equation for y. -3(0) + 2 y = 12 2 y= 12 2 2 y = 6 or (0, 6) 3/16/2018 Smyrna High School Geometry CP 46

How Do We Graph a Linear Equation? ? ? You try… 6 x - 4 y = 12 3/16/2018 Smyrna High School Geometry CP 47

Time to practice… • Let’s do #5 & 6 together • Now it is your turn to finish the odd problems (1, 3, 7) Make sure that you have me check these problems (10 mins) • If you finish early, go ahead and finish the even problems (2, 4, 8). • If you don’t finish the evens in class, you can finish them for homework. graphing lines using intercepts. pdf 3/16/2018 Smyrna High School Geometry CP 48

How Do We Graph a Linear Equation? ? ? Let’s Practice with Hamilton High: 20 x + 10 y = 240 • Find the x-intercept: Let y = 0 and solve the equation for x. 20 x + 10(0) = 240 20 x = 240 20 20 x = 12 or (12, 0) 3/16/2018 Smyrna High School Geometry CP 49

How Do We Graph a Linear Equation? ? ? Let’s Practice: 20 x + 10 y = 240 Find the y-intercept: Let x = 0 and solve the equation for y. 20(0) + 10 y = 240 10 x = 240 10 10 y = 24 or (0, 24) 3/16/2018 Smyrna High School Geometry CP 50

How Do We Graph a Linear Equation? ? ? Let’s Practice: x + y = 18 • Find the x-intercept: Let y = 0 and solve the equation for x. x + (0) = 18 x = 18 or (18, 0) 3/16/2018 Smyrna High School Geometry CP 51

How Do We Graph a Linear Equation? ? ? Let’s Practice: x + y = 18 Find the y-intercept: Let x = 0 and solve the equation for y. (0) + y = 18 or (0, 18) 3/16/2018 Smyrna High School Geometry CP 52

Putting It All Together ! ! Complete part f from the Hamilton High problem. Plot your intercepts to graph both lines. What is your solution to the system? 3/16/2018 Smyrna High School Geometry CP 53

Let’s verify • How can we verify that (6, 12) is the solution to the system of equation: 20 x + 10 y = 240 x + y = 18 3/16/2018 • You must verify the solution into BOTH equations for x AND y. 20 x + 10 y = 240 20(6) + 10(12) = 240 120 + 120 = 240 = 240 x + y = 18 6 + 12 = 18 18 = 18 Smyrna High School Geometry CP 54

How do you verify solutions? (Is it the answer? ) • Determine whether the point (3, 8) is a solution to each system of equations. 2 x + y = 14 x + y = 11 3/16/2018 y = -x – 5 y=x+5 4 x – y = - 4 3 x – 2 y = 7 Smyrna High School Geometry CP 55

Verifying Solutions • Determine whether the point (3, 8) is a solution to each system of equations. 2 x + y = 14 x + y = 11 y = -x – 5 y=x+5 4 x – y = - 4 3 x – 2 y = 7 2(3) + 8 = 14 (YES) 3 + 8 = 11 (YES) 8 = -3 – 5 (NO) 8 = 3 + 5 (YES) 4(3) - 8 = - 4 (NO) 3(3) -2(8) = 7 (NO) YES! Both equations are true NO! Only 1 equation is true NO! Neither equations are true 3/16/2018 Smyrna High School Geometry CP 56

Practice Makes Perfect! On the whiteboard, solve the system graphically (use x & y intercepts to graph) and then verify the solution. Odd desk – do #1 Even desk – do #2 1. 2 x + y = 8 2. -3 x + 2 y = 6 -x + y = 2 3 x + y = 12 Bonus! 3/16/2018 7 x + 2 y = 5 2 x + 3 y = 16 57

Think-Pair-Share A Better Deal When the date for the work project was set, it turned out that only 13 science club members could participate. The club president talked again with the PTA president and got a new pay deal $20 per outdoor worker and $15 per indoor worker. Still, $240 is needed for the club… 58

REMIND YOUR GROUPIES Left: How do I know if my answer is correct? Middle: What is a solution to a system of equations? Right: How do I graph an equation if it is not in y = form? 3/16/2018 Smyrna High School Geometry CP 59

Summarize 3 -2 -1 3/16/2018 Smyrna High School Geometry CP 60

Summarize! Journal Time How can you solve a system of equations graphically if it is not in slope- intercept form? What are the steps? 3/16/2018 Smyrna High School Geometry CP 61

Common Core Standards • A. CED. 1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. • A. CED. 2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. • F. BF. 1 Write a function that describes a relationship between two quantities. ★ • a. Determine an explicit expression, a recursive process, or steps for calculation from a context. 3/16/2018 Smyrna High School Geometry CP 62