MJ2A

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Solve for y, given the value of x
Solutions y = x + 1 x = 2 y = x – 4 x = -1 x + y = -1 x + y = 0 x = 1 x = 4
3
-1
-2
-4

 Text p. 407 – 408 # 15 – 25

 Please take out your notebook and get ready to work…
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In the last lesson we worked with writing linear equations using the slopeintercept form.
In today’s lesson we will look at pairs of linear equations and their common solutions…

 Students will solve systems of linear equations by the graphing & substitution methods

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Systems of Equations are sets of linear equations.
Solution – is the common ordered pair that make the system of equations true.

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There are several methods to solving systems of equations.
In today’s lesson we will look at
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Solving by graphing
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Solving by substitution

 When solving systems of equations by graphing you will be presented with or asked to graph 2 linear equations in the same coordinate plane.
 When analyzing the graphs there are three possible scenarios as follows:
 One Solution
 No Solution
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 Infinite Solutions
Let’s see what they look like…

 When analyzing a graph with systems of equations, the lines will intersect.
 The intersection point represents the solution of both equations
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You can check the solution by substituting the ordered pair into both equations and you should get a true statement for both equations
Let’s look at an example…

 This is the graph of the equations y = -x y = x + 2
 Notice that the lines intersect at the point (-1, 1)
 The ordered pair represents the solution to the system of equations x (-1, 1) y

 You can check to make sure that the ordered pair
(-1, 1) is the solution by substituting the x and yvalues into the original equations as follows:
Equation #1 Equation #2 y = -x
1 = - (-1)
1 = 1
 y = x + 2
1 = -1 + 2
1 = 1


 Observe that the slope of each equation is different.
y = -x m = -1 y = x + 2 m = 1
 Without doing any work, we can analyze the slopes of the equations and predict how many solutions there will be to a system of equations
 If the slopes are different , then there will be only one solution to the system of equations

 In a graph with 2 parallel lines there will be no solution to the system of equations
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In this instance, the lines do not intersect.
Therefore, no ordered pair will be the solution to both equations…
Let’s see what that looks like…

 This is a graph of the equations y = 2x + 4 y = 2x - 1
Notice that the lines are parallel and do not intersect x
Because the lines do not intersect there is no solution to this system of equations y

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Observe the slope an y-intercepts of each equation.
y = 2x + 4 m = 2, b = 4 y = 2x - 1 m = 2, b = -1
Without doing any work, we can analyze the slopes and y-intercepts of the equations and predict how many solutions there will be to the system of equations
If the slopes are same , with different y-intercepts then there will be no solution to the system of equations

 In a system of equations where the graph of each line is the same you will have an infinite number of solutions.
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Because the graphs of each line overlap, all points on each line will make the system of equations true, therefore you will have an infinite number of solutions
Let’s look at an example…

 This is a graph of the equations
2y = x + 6 y = ½ x + 3 x Both equations have the same graph
Any ordered pair on the lines will make both equations true y

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Observe that the slope an y-intercepts of each equation.
2y = x + 6 y = ½ x + 3 m = ½ , b = 3 m = ½, b = 3
If you transform the first equation into the slopeintercept form you will see that both equations have the same slope and y-intercept
If the slopes and y-intercepts are the same then there will be an infinite number of solutions to the system of equations

 You can also solve a system of equations by the substitution method.
 First, you need to know that you cannot solve an equation with two variables.
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In the substitution method you substitute the value of one variable into the other equation…
Let’s see what that looks like…

Solve y = x – 5 y = 3
In this instance the second equations gives you the value of y.
To solve, substitute the value of y into the first equation and solve algebraically as follows: y = x – 5
3 = x – 5
+5 + 5
8 = x
The solution to the system of equation is x = 8 and y = 3
The ordered pair that will make this system of equation true is (8, 3)

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In the notes section of your notebook write and solve the system of equations using the substitution method.
Solutions y = 3x – 4
(0, -4) x = 0 x + y = 8 (2, 6) y = 6

 In the notes section of your notebook summarize the key concepts covered in today’s lesson
 Today we discussed:
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Solving systems of equations by graphing – what are the solution scenarios?
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Solving systems of equations by substituting – how does that work?

 Text p. 417 # 12 – 23
Reminder
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This assignment is due tomorrow
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I do not accept late assignments
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You must show how you got your answers or no credit (no work = no credit)