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 – 3-2-1 S – 3-2-1 Homework: Graphing Equations worksheet #1-6 1 06/27/22 Smyrna High School Geometry CP
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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 B Review: Graph linear equations.
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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 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.
• 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.
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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
504/19/23 Smyrna High School Geometry CP
How to Graph an Equation in
Slope-Intercept Form
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).
Check Skills You Will NeedCheck Skills You Will Need
04/19/23 6Smyrna High School Geometry CP
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.
Check Skills You Will NeedCheck Skills You Will Need
04/19/23 7Smyrna High School Geometry CP
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.
Check Skills You Will NeedCheck Skills You Will Need
04/19/23 8Smyrna High School Geometry CP
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 find a third point.
Check Skills You Will NeedCheck Skills You Will Need
904/19/23 Smyrna High School Geometry CP
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.
Check Skills You Will NeedCheck Skills You Will Need
04/19/23 10Smyrna High School Geometry CP
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 lower on the right side of the graph.
Check Skills You Will NeedCheck Skills You Will Need
1104/19/23 Smyrna High School Geometry CP
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.04/19/23 Smyrna High School
Geometry CP 12
How to Graph an Equation in
Slope-Intercept Form
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
Check Skills You Will NeedCheck Skills You Will Need
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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04/19/23 Smyrna High School Geometry CP
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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.
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Systems of Equations: Example 1Systems of Equations: Example 1
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Smyrna High School Geometry CP
Systems of Equations: Example Systems of Equations: Example 22
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Systems of Equations: Example Systems of Equations: Example 33
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3 Methods to Solve 3 Methods to Solve System of EquationsSystem of Equations
3 Possible Solution 3 Possible Solution TypesTypes
• 1 Exact Solution
• No Solution
• Infinitely Many Solutions
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1 Exact Solution1 Exact Solution
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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
Smyrna High School Geometry CP
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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.
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Hamilton High SchoolHamilton High School
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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
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Hamilton High SchoolHamilton High School
• What does the x represent?
x: # of outdoor workers
• What does the y represent?
• y: # of indoor workers
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Hamilton High SchoolHamilton High School
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
04/19/23 Smyrna High School Geometry CP
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Hamilton High SchoolHamilton High School
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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04/19/23 Smyrna High School Geometry CP
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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
4004/19/23 Smyrna High School Geometry CP
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.
04/19/23 Smyrna High School Geometry CP
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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.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
04/19/23 43Smyrna High School Geometry CP
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
How Do We Graph a How Do We Graph a Linear Equation???Linear Equation???
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)
04/19/23 45Smyrna High School Geometry CP
How Do We Graph a How Do We Graph a Linear Equation???Linear Equation???
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)
04/19/23 46Smyrna High School Geometry CP
How Do We Graph a How Do We Graph a Linear Equation???Linear Equation???
You try…
6x - 4y = 12
04/19/23 47Smyrna High School Geometry CP
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
How Do We Graph a How Do We Graph a Linear Equation???Linear Equation???
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
How Do We Graph a How Do We Graph a Linear Equation???Linear Equation???
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
How Do We Graph a How Do We Graph a Linear Equation???Linear Equation???
Let’s Practice:
x + y = 18
04/19/23 51Smyrna High School Geometry CP
•Find the x-intercept: Let y = 0 and solve the equation for x.
x + (0) = 18x = 18
x = 18 or (18, 0)
How Do We Graph a How Do We Graph a Linear Equation???Linear Equation???
Let’s Practice:
x + y = 18
04/19/23 52Smyrna High School Geometry CP
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
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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
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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?
How can you solve a system of equations graphically if it is not in
slope- intercept form? What are the steps?
04/19/23 61Smyrna High School Geometry CP
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.