Today’s Agenda E -E - How can graphs be used to solve systems of linear equations? A -A - Warm-up: Bell Ringer Patterns B Review: Graph linear equations

Today’s Agenda• E - E - How can graphs be used to solve systems of linear

equations?

• A - A - Warm-up: Bell Ringer Patterns B

• Review: Graph linear equations from slope-intercept form (graphic organizer)

• T - T - New: System of Linear Equations

(graphic organizer and video)– Class work: A Good Deal Shoulder Buddy Practice

Problem

• S – 3-2-1S – 3-2-1Homework: Graphing Equations worksheet #1-6

104/19/23 Smyrna High School Geometry CP

Warm-upWarm-up

Bell Ringer Day 1~Discovering Patterns…•Complete the pattern with your group. Then answer the questions as best you can.


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http://www.google.com/url?sa=i&rct=j&q=warm-up&source=images&cd=&cad=rja&docid=scmbEyY9hLmJXM&tbnid=cVmP82Ah1JRLHM:&ved=0CAUQjRw&url=http://www.essentialtennis.com/podcast/warming-up/ultimate-tennis-warmup-guide-podcast-213/16602/&ei=p2n8UbD-Juri4AP-soDoCw&bvm=bv.50165853,d.dmg&psig=AFQjCNFE5bdTfIV3dnqAA6lHZ96tjB0rdg&ust=1375583014242318

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?

• Write a rule for this pattern?04/19/23 Smyrna High School Geometry CP

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Unit 1 – Lesson 3: Systems of Linear

Equations

Systems of EquationsSystems of Equations

Two or more linear equations that have to be

solved simultaneously.

04/19/23 4Smyrna High School Geometry CP

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-intercept form can be graphed.

Check Skills You Will NeedCheck Skills You Will Need




y = mx + b

• With these two facts, lets look at some examples.

• y = 4x + 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 (0, b).





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.





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.





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 find a third point.





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.





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 lower on the right side of the graph.



Pattern C

• How many toothpicks are in figure 5?



• Write a rule for this pattern.04/19/23 Smyrna High School

Geometry CP 12



y = mx + b

• For practice, graph the following equations in slope-intercept form.

1.y = -x + 6

2.y = 2x – 4

3.y = ¼ x




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(0, 6) -1/1

(0, - 4) 2/1

(0, 0) 1/4

Graph!

• http://www.youtube.com/watch?v=2BHzXItkByU


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http://www.youtube.com/watch?v=2BHzXItkByU

http://www.youtube.com/watch?v=2BHzXItkByU

Ticket In The Door RecapTicket 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!


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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


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Systems of EquationsSystems of Equations

• Let’s take a look at a few examples.

• Your task: I Notice, I WonderI Notice, I Wonder.– Discuss and write down some things

that you NOTICENOTICE about the examples.

– Discuss and write down some things that you WONDERWONDER about the examples.


Systems of Equations: Example 1Systems of Equations: Example 1

04/19/23 20

Smyrna High School Geometry CP

Systems of Equations: Example Systems of Equations: Example 22


Systems of Equations: Example Systems of Equations: Example 33


3 Methods to Solve 3 Methods to Solve System of EquationsSystem of Equations

• Graphing Method• Substitution Method• Elimination Method


3 Possible Solution 3 Possible Solution TypesTypes

• 1 Exact Solution

• No Solution

• Infinitely Many Solutions


1 Exact Solution1 Exact Solution


Point of intersection


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Another type of solutionAnother type of solution

• How would you describe these lines?

Y = 3x + 2

Y = 3x - 4

What do you think the solution (point of intersection) is?04/19/23

No SolutionNo SolutionParallel LinesWill not intersect

same same slope slope

different y-different y-interceptsintercepts04/19/23 Smyrna High School

Geometry CP 27


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Another type of solutionAnother type of solution

• What do you notice about the graphs and equations?

Y = -3x + 4

3x + y = 4

What do you think the solution (point of intersection) is?

04/19/23

Infinitely Many SolutionsInfinitely Many Solutions

SAME LINE

They are the They are the SAMESAME equation equation

(just written in different forms)(just written in different forms)

They have They have same slope same slope and and same y-interceptsame y-intercept..


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The Goal of Solving The Goal of Solving SystemsSystems

• To find one pair (x, y) of values that satisfies both linear equations.–The one pair of values that makes

both equations true.


Hamilton High SchoolHamilton High School


Hamilton High School

• Students in the Hamilton High School science club want to raise $240 to buy a special eyepiece for the high-powered 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.


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Hamilton High School

• What do the variables x and y represent in these equations?

• What problem condition is represented by the equation

20x + 10y = 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?


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Think-Pair-ShareThink-Pair-ShareLet’s Look at the Let’s Look at the

SolutionSolution

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



• What does the x represent?

x: # of outdoor workers

• What does the y represent?

• y: # of indoor workers


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20x + 10y = 24020x + 10y = 240

• What does this equation represent in the problem?

• 20x + 10y = 240 shows the amount of money that can be earned depending on the # of outdoor and indoor workers


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x + y = 18x + y = 18

• What does this equation represent in the problem?

• x + y = 18 shows that the # of club members who will work


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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?


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Today’s Agenda• E - E - How can graphs be used to solve systems of linear

equations?

• A - A - Warm-up: Bell Ringer Patterns

• Review: Graph linear equations from slope-intercept form (graphic organizer)

• T - T - New: Graph linear equations from intercepts

(graphic organizer)– Class work: A Good Deal Shoulder Buddy Practice

Problem, Verifying solutions, Solving systems

• S – S – Journal entryJournal entryHomework: Pattern D


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.


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HOMEWORK: Pattern D




• Write a rule for this pattern.04/19/23 Smyrna High School

Geometry CP 42

How Do We Graph a How Do We Graph a Linear Equation???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


How Do We Graph a How Do We Graph a Linear Equation not in Linear Equation not in

y=mx+b form???y=mx+b form???Let’s Practice:

-3x + 2y = 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.04/19/23 44Smyrna High School

Geometry CP


Let’s Practice:

-3x + 2y = 12

•Find the x-intercept: Let y = 0 and solve the equation for x.

-3x + 2(0) = 12-3x = 12

-3 -3x = -4 or (-4, 0)



Let’s Practice:

-3x + 2y = 12

•Find the y-intercept: Let x = 0 and solve the equation for y.

-3(0) + 2y = 122y= 12

2 2y = 6 or (0, 6)



You try…

6x - 4y = 12


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.pdf04/19/23 Smyrna High School

Geometry CP 48


Let’s Practice with Hamilton High:

20x + 10y = 240•Find the x-intercept: Let y = 0 and solve the equation for x.

20x + 10(0) = 24020x = 24020 20

x = 12 or (12, 0)04/19/23 49Smyrna High School

Geometry CP


Let’s Practice:

20x + 10y = 240

Find the y-intercept: Let x = 0 and solve the equation for y.

20(0) + 10y = 24010x = 24010 10

y = 24 or (0, 24)04/19/23 50Smyrna High School

Geometry CP


Let’s Practice:

x + y = 18


•Find the x-intercept: Let y = 0 and solve the equation for x.

x + (0) = 18x = 18

x = 18 or (18, 0)


Let’s Practice:

x + y = 18


Find the y-intercept: Let x = 0 and solve the equation for y.

(0) + y = 18y = 18

y = 18 or (0, 18)

Putting It All Together !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?


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Let’s verifyLet’s verify

• How can we verify that (6, 12) is the solution to the system of equation:

20x + 10y = 240

x + y = 18

• You must verify the solution into BOTH equations for x AND y.

20x + 10y = 240

20(6) + 10(12) = 240

120 + 120 = 240

240 = 240

x + y = 18

6 + 12 = 18

18 = 18Smyrna High School Geometry CP

04/19/23 54

How do you verify How do you verify solutions?solutions?

(Is it the answer?) (Is it the answer?)

04/19/23

Smyrna High School Geometry CP 55

• Determine whether the point (3,8) is a solution to each system of equations.

2x + y = 14 x + y = 11

y = -x – 5y = x + 5

4x – y = - 43x – 2y = 7

Verifying SolutionsVerifying Solutions

04/19/23Smyrna High School Geometry CP 56

• Determine whether the point (3,8) is a solution to each system of equations.

2x + y = 14 x + y = 11

2(3) + 8 = 14 (YES) 3 + 8 = 11 (YES)

YES! Both equations are true

y = -x – 5y = x + 5

8 = -3 – 5 (NO)8 = 3 + 5 (YES)

NO! Only 1 equation is true

4x – y = - 43x – 2y = 7

4(3) - 8 = - 4 (NO)3(3) -2(8) = 7 (NO)

NO! Neither equations are true

Practice Makes Perfect!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

1.2x + y = 8 2. -3x + 2y = 6

-x + y = 2 3x + y = 12

Bonus! 7x + 2y = 5

2x + 3y = 16

04/19/23 57

Think-Pair-ShareA 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…

04/19/23

58

Smyrna High School

Geometry CP

REMIND YOUR GROUPIESREMIND YOUR GROUPIES


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Right: How do I graph an equation if it is not in y = form?

Left: How do I know if my answer is correct?

04/19/23

Middle: What is a solution to a system of equations?

Summarize Summarize 3-2-1 3-2-1


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Methods to Solve Systems of Equations

Items to Identify when Graphing a Linear Equation

Way to Identify the Solution of a Graphed System

Name: ___________________________ Date: ____________________________

Summarize!Summarize! Journal Time Journal Time

How can you solve a system of equations graphically if it is not in

slope- intercept form? What are the steps?


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.


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Documents

Today’s Agenda E -E - How can graphs be used to solve systems of linear equations? A -A - Warm-up: Bell Ringer Patterns B Review: Graph linear equations