Chapter 3 Systems of Linear Equations and Inequalities

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Chapter 3Systems of Linear Equations and
Inequalities
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Section 3.1Solving Linear Systems by Graphing
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Definitions
We previously learned how to graph linear
equations in slope-intercept form
y mx b
Where m represents the slope b
represents the y-intercept
We plotted the y-intercept then used the slope to
find a second point.
Solutions to a linear equation are represented by
all points on the line of its graph.
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Solutions to Linear Equations
Determine whether the given ordered pair (x,y)
is a solution for the specified linear equation
Ordered Pair
(5,2)
Linear Equation
3x 2y 11
Substitute the values of the ordered pair into
the equation.
3x 2y 11
3(5) 2(2) 11
15 4 11
11 11
Yes. It is a solution.
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Solutions to Linear Equations
Determine whether the given ordered pair (x,y)
is a solution for the specified linear equation
Ordered Pair
(5,2)
Linear Equation
-x 6y 7
Substitute the values of the ordered pair into
the equation.
-x 6y 7
- (5) 6(2) 7
-5 12 7
7 7
Yes. It is a solution.
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Solutions to Linear Equations
So, the ordered pair (5,2) is a solution for
both the linear equation 3x 2y 11 and -x
6y 7.
How can you determine this graphically?
Lets graph both equations.
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Graphing Linear Equations
Graph the following linear equation
Find the intercepts
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Graphing Linear Equations
Now, graph the following linear equation
Find the intercepts
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Graphing Linear Equations
Lets combine the graphs
and
Where do they intersect?
(5,2)
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Solutions to Linear Equations
In the previous example, the two equations
represent a system of linear equations 3x
2y 11 -x 6y 7
A solution of a system of linear equations in two
variables is an ordered pair (x, y) that
satisfies each equation in the system.
In this example, the point (5,2) is the only
solution as illustrated on the graph.
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Solutions to Linear Equations
There are three possible scenarios for the
solutions to a system of equations.
Infinite of Solutions
One Solution
No Solution
One Intersection Point
Parallel Lines
Coinciding Lines (Graphs of Equations Represent
the Same Line)
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Example 1
Find the solution to the following system of
equations
and
Where do they intersect?
(3,-3)
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Example 1 (continued)
You can check the answer by substituting the
ordered pair into each equation.
Solution
(3, -3)
y 3x - 12
y -2x 3
(-3) 3(3) - 12
(-3) -2(3) 3
-3 9 - 12
-3 -6 3
-3 -3
-3 -3
Solution Checks
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Example 2
Find the solution to the following system of
equations
and
First solve for y.
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Example 2 (continued)
Next, graph the equations
and
Where do they intersect?
(4,-1)
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Example 2 (continued)
You can check the answer you arrived at
graphically by substituting the ordered pair into
each equation.
Solution
(4, -1)
y -3x 11
y ½x - 3
(-1) -3(4) 11
(-1) ½(4) - 3
-1 -12 11
-1 2 - 3
-1 -1
-1 -1
Solution Checks
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Example 3
Find the solution to the following system of
equations
and
First find the intercepts
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Example 3 (continued)
Next, graph the equations
and
Where do they intersect?
(-1,0)
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Example 3 (continued)
Check the answer you arrived at graphically by
substituting the ordered pair into each equation.
Solution
(-1, 0)
y 2x 2
y - x - 1
(0) 2(-1) 2
(0) - (-1) - 1
0 -2 2
0 1 - 1
0 0
0 0
Solution Checks
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Example 4
Find the solution to the following system of
equations
and
First solve for y.
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Example 4 (continued)
The equations represent the same line
and
Where do they intersect?
Infinite of Points
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Example 5
Find the solution to the following system of
equations
and
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Example 6 (continued)
Next, graph the equations
and
Where do they intersect?
Parallel Lines (No Solution)