unit 1 – lesson 3: systems of linear equations systems of equations two or more linear equations...
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Unit 1 – Lesson 3: Unit 1 – Lesson 3: Systems of Linear EquationsSystems of Linear Equations
Systems of Equations
Two or more linear equations that have to be solved at the same
time.
Systems of EquationsSystems 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.
– Discuss and write down some things that you WONDER about the examples.
Systems of Equations: Example 1Systems of Equations: Example 1
Systems of Equations: Example 2Systems of Equations: Example 2
Systems of Equations: Example 3Systems of Equations: Example 3
Geometry Honors 6
How do we “solve” a system How do we “solve” a system of equations??? of equations???
• By finding the point where two or more equations, intersect. This is called the SOLUTION.
x + y = 6x + y = 6
y = 2xy = 2x Point of intersectionPoint of intersection
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““All I do is Solve” videoAll I do is Solve” video
• http://www.youtube.com/watch?v=1qHTmxlaZWQ&feature=related
3 Methods to Solve 3 Methods to Solve System of EquationsSystem of Equations
• Graphing• Substitution Method• Elimination Method
3 Possible Solution Types3 Possible Solution Types
• 1 Solution• No Solution• Infinitely Many Solutions
• http://www.algebra-class.com/graphing-systems-of-equations.html
1 Solution1 Solution
Point of intersection
Geometry Honors 11
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 Solution
Parallel LinesWill not intersect
Geometry Honors 13
PARALLEL LINESPARALLEL LINES
No SolutionNo Solution: : • when lines of a graph are parallelwhen lines of a graph are parallel
• Parallel lines have the same slope but different y-interceptsParallel lines have the same slope but different y-intercepts
• since they do not intersect, there is no solutionsince they do not intersect, there is no solution
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Geometry Honors 14
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?
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Infinitely Many SolutionsInfinitely Many Solutions
SAME LINE
Geometry Honors 16
Infinite SolutionsInfinite Solutions: :
INFINITELY MANY INFINITELY MANY SOLUTIONSSOLUTIONS
• a pair of equations that have the same a pair of equations that have the same slope and same y-intercept.slope and same y-intercept.
• They are the They are the SAMESAME equation (just written in equation (just written in different forms)different forms)
•Since they are the Since they are the SAME EQUATIONSAME EQUATION, they , they have the have the SAME LINESAME LINE
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Geometry Honors 17
Does it have a solution?Does it have a solution?
1) 1)
Determine whether the following have one, none, or infinite Determine whether the following have one, none, or infinite solutions by identifying the solutions by identifying the slopeslope and and y-intercepty-intercept. . Explain your reasoning. Explain your reasoning.
2y = 8 - x2y = 8 - x
y = 2x + 4y = 2x + 4
3) 3) 2) 2) x - 5y = 10x - 5y = 10
-5y = -x +6-5y = -x +6
y = -6x + 8y = -6x + 8
y + 6x = 8y + 6x = 8
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Geometry Honors 18
Does it have a solution?Does it have a solution?
1) 1)
Determine whether the following have one, none, or infinite Determine whether the following have one, none, or infinite solutions by just looking at the solutions by just looking at the slopeslope and and y-interceptsy-intercepts
2y + x = 82y + x = 8
y = 2x + 4y = 2x + 4
3) 3) 2) 2) x - 5y = 10x - 5y = 10
-5y = -x +6-5y = -x +6
y = -6x + 8y = -6x + 8
y + 6x = 8y + 6x = 8
ANS:ANS: One SolutionOne Solution
ANS:ANS: No Solution
ANS:ANS: Infinite Solutions
04/19/23
Systems of Equations VideoSystems of Equations Video
• Systems of Equations: Part 01–Watch carefully as this video
explains what a system of equations are and gives a fantastic real-world example of how systems are used in the business world.
The Goal of Solving SystemsThe 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.
Hamilton High SchoolHamilton High School
Hamilton High SchoolHamilton High School
16x + 10y = 24016x + 10y = 240
x + y = 18x + y = 18
• What does the x represent?
x: # of outdoor workers
• What does the y represent?
• y: # of indoor workers
Hamilton High SchoolHamilton High School
16x + 10y = 24016x + 10y = 240
• What does this equation represent in the problem?
• 16x + 10y = 240 shows the amount of money that can be earned depending on the # of outdoor and indoor workers
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
Hamilton High SchoolHamilton High School
16x + 10y = 24016x + 10y = 240
• Determine three combination of outdoor (x) and indoor (y) workers so that the club earns exactly $240.
• SHOW ALL WORK!!
Hamilton High SchoolHamilton High School
x + y = 18x + y = 18
• Do any of the combinations from part d work for the 18 workers that are needed?
• SHOW ALL WORK!!
Let’s verifyLet’s verify
• How can we verify that (10,8) is the solution to the system of equation:
16x + 10y = 240
x + y = 18
• You must verify the solution into BOTH equations for x AND y.
16(10) + 10(8) = 240
160 + 80 = 240
240 = 240
10 + 8 = 18
18 = 18
Geometry CP
Where’s the solution?Where’s the solution?
• Use the graph to estimate a solution for the system of equations (basically what x and y value works for BOTH equations)
• SOLUTION: POINT OF INTERSECTION (10, 8)
SOLUTION
Hamilton High SchoolHamilton High School
• Plugging and chugging numbers is exhausting and very time consuming.
• What other strategies could you use to find a pair of values (x,y) that satisfy BOTH equations at the same time?
How Do We Graph a Linear How Do We Graph a Linear Equation???Equation???
In order to graph a linear equation it HAS to be
in the form y = mx + b,
where m is the slope and b is the y-intercept
How Do We Graph a Linear How Do We Graph a Linear Equation???Equation???
Let’s Practice:
16x + 10y = 240
How Do We Graph a Linear How Do We Graph a Linear Equation???Equation???
Let’s Practice:
x + y = 18
Let’s Look at the SolutionLet’s Look at the Solution
Complete Problem 1 - Parts a, b, & c.
A Better Deal 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.
MATCHING ACTIVITYMATCHING ACTIVITY
Verifying SolutionsVerifying Solutions
04/19/23 Geometry Honors 36
• 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
04/19/23 Geometry Honors 37
• 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
REMIND YOUR SHOULDER BUDDY
Geometry Honors 38
What do you do if the equation is not in y= form?
How do I know if my answer is correct?
04/19/23
REMIND YOUR SHOULDER BUDDY
Geometry Honors 39
Always replace x and y for BOTH equations to verify your solution.
What do you do if the equation is not in y= form?
You have to rewrite it solving for y so that it can be graphed.
How do I know if my answer is correct?
04/19/23
Geometry Honors 40
GRAPHING CALCULATORGRAPHING CALCULATOR
• Rewrite equation in y = form
• Use the INTERSECT function to find the intersection point
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Geometry Honors 41
GRAPHING CALCULATOR EXAMPLES
• y = - 3x and 4x + y = 2
• x + y = 1 and 2x + y = 4
• 3x + y = 1 and –x + 2y = 16
• 2x + y = 1 and 5x + 4y = 10
04/19/23
Geometry Honors 42
GRAPHING CALCULATOR EXAMPLES
• y = - 3x and 4x + y = 2
• x + y = 1 and 2x + y = 4
• 3x + y = 1 and –x + 2y = 16
• 2x + y = 1 and 5x + 4y = 10
04/19/23
(2, - 6)
(3, -2)
(-2, 7)
(-2, 5)
Geometry Honors 43
GRAPHING CALCULATOR EXAMPLES
• y = 3x + 2 and y = 3x – 4
• y = -3x + 4 and 3x + y = 4
04/19/23
Geometry Honors 44
GRAPHING CALCULATOR EXAMPLES
• y = 3x + 2 and y = 3x – 4
• y = -3x + 4 and 3x + y = 4
04/19/23
Infinitely many
No solution
Daily Homework Quiz For use after Lesson 7.1
Use the graph to solve the linear system1.
3x – y = 5
–x + 3y = 5
ANSWER (2, 1)
Daily Homework Quiz For use after Lesson 7.1
2. Solve the linear system by graphing.
2x + y = –3 –6x + 3y = 3
ANSWER (–1, –1)
SummarizeSummarize
3 – 2 – 1 3 methods to solve systems of
equations
2 important items to identify when graphing a linear equation
1 way to identify the solution of a graphed systems of equations