Download Solving Linear Systems: Graphing Method

Solving Linear Systems of Equations - Graphing Method
• Recall that to solve the
linear system of equations
in two variables ...
a1x  b1 y  c1
a 2 x  b2 y  c2
we need to find the value of x and y that satisfies both
equations. In this presentation the Graphing Method
will be demonstrated.
xy 4
x  y  2
• Consider the system of
equations ...
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Solving Linear Systems of Equations - Graphing Method
• Sketch a graph of the first
equation ...
• Each point on the graph
of the function satisfies the
equation of the function ...
xy 4
 yx4
4
• (x,y)
2
-4
-2
0
2
4
-2
-4
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Slide 2
Solving Linear Systems of Equations - Graphing Method
• Now add a sketch of the
second function to the
graph.
x  y  2
 y x2
• Each point on the graph
of this function satisfies the
equation of the function ...
4
2
-4
-2
• (x,y)
0
2
4
-2
-4
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Slide 3
Solving Linear Systems of Equations - Graphing Method
• Now consider the point of intersection of the two
graphs ...
• Since this point is on both graphs, it must be a solution
point for both equations!
• We only need to
determine the coordinates
of the point to have the
solution to the system of
equations.
4
(1,3)
2
-4
-2
0
2
4
-2
• It appears that the point is ...
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-4
Slide 4
Solving Linear Systems of Equations - Graphing Method
• Let x = 1 and y = 3 in each equation.
x  y  2
1  3  2
 2  2
xy4
1 3  4
44
• The ordered pair (1, 3) is
now confirmed as the
solution to the system of
equations.
4
(1,3)
2
-4
-2
0
2
4
-2
-4
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Slide 5
Solving Linear Systems of Equations - Graphing Method
• Summary:
1) Graph both functions on the same coordinate
plane.
2) Determine the point of intersection for the two
graphs.
3) Substitute the coordinates of the point into
both equations to make sure it is a solution
to the system.
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Slide 6
Solving Linear Systems of Equations - Graphing Method
• While the graphing method is helpful in gaining a
visual understanding of solving systems of equations, it
is not very practical for solving systems whose solutions
are not integers. For example, the solution (3/7, - 2/9) is
not likely to be found by this method.
• When approximate solutions are desired, the graphing
calculator can be used very effectively with this method.
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Slide 7
Solving Linear Systems of Equations - Graphing Method
• Example: use a graphing calculator to find the
solution to the following system (nearest hundredth).
3x  4 y  7
2x  3y  2
• Place each equation into y = f(x) form:
3x  4 y  7
4 y  3x  7
3
7
y x
4
4
2 x  3 y  2
 3y  2 x  2
2
2
y x
3
3
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Slide 8
Solving Linear Systems of Equations - Graphing Method
3
7
y x
4
4
• Enter the functions into
Y1 and Y2 on the
calculator ...
2
2
y x
3
3
• Use a ZOOM|ZStandard
window ...
• Use CALC|intersect to
find the point of
intersection ...
• The solution is
x  0.76
y  1.18
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Slide 9
Solving Linear Systems of Equations - Graphing Method
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