Systems

1
Solving systems of equations and inequalities by
graphing
2
Systems of equations

• Remember that a system of equations is a group of
  two or more equations that we solve at the same
  time
• A point is a solution of the system if it works
  when substituted into each equation. For
  example, the solution to the system above is
  (2,0).

3
Review of graphs of systems of linear equations

• When working with two equations in two variables,
  there are three possibilities for their graphs

The lines can intersect and have one solution (x,
y).
The lines can be parallel and have no solution.
The lines can coincide and have infinitely many
solutions.
4
But now

• We want to start working with systems that dont
  just have linear equations.
• We will still graph our functions and look for
  the point(s) of intersection when we want to
  solve our systems.

5
EXAMPLE 1

• Lets solve the system below by graphing

• Graph each function on the same coordinate plane

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EXAMPLE 1 CONTINUED

• Look at the graph and identify the points of
  intersection

There are two points of intersection, so our
system has two solutions (-1, 2) and (1,
2) You can substitute both points into your
equations and get true statements. This is an
easy way to check your work!
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To solve using your calculator

• Put your equations in y . abs( can be found by
  pressing 2nd 0, and choosing the first option.
• Graph to see the number of solutions.

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To solve using your calculatorcontinued

• To find the first point of intersection, press
  2nd TRACE, and choose 5 (intersect). Move your
  cursor to the left of the first intersection and
  press enter. Move to the right and press enter.
  Then press enter a third time to see the
  coordinates
• Repeat the process to find the second solution at
  (1, 2).

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EXAMPLE 2

• Lets solve
• First, recognize that the first equation is an
  absolute value graph (a V) that has been shifted
  right 2 units and down 1 unit.
• Then, solve the second equation for y y x
  1.
• Finally, graph.

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Example 2 continued

• The graphs intersect ONCE.
• The only solution to the system is (0, 1).
• Notice that you can substitute your point into
  both equations and get a true statement.

11
EXAMPLE 3

• Lets solve
• First, solve the first equation for y to get
• . Then, recognize that
  this is an absolute value graph (a V) that has
  been shifted left 2 units, down 2 units, and
  reflected across the x-axis.
• The second equation is a line.
• Now, graph.

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Example 3 continued

• The graphs dont intersect.
• The solution is that there is no solution.
• This means there is NO point that exists that
  would give you a true statement for both
  equations.

13
Summary of steps

• Graph each function in your system. It would be
  most helpful if you solve for y in each case.
• Identify the point(s) of intersection of the
  graphs of your functions.
• State your solution(s). Check them by
  substituting back into your system of equations.

14
Systems of INEQUALITIES

• Remember that a system of inequalities is a group
  of two or more equations that we solve at the
  same time
• Heres a review of what the symbols tell us to
  do
• gt dashed line, shaded above boundary line
• lt dashed line, shaded below boundary line
• solid line, shaded above boundary line
• solid line, shaded above boundary line

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Systems of Inequalities Continued

• We will graph each boundary line just as we did
  before, and we will put each of them on the same
  coordinate plane.
• Where the shaded regions all overlap will
  represent the solution of our systemmeaning that
  any point from the shared region will produce a
  true solution when substituted into all of the
  inequalities in our system

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EXAMPLE 1

• Lets solve the system below by graphing
• The first will be a dashed line shaded above. (in
  red)
• The second will be a solid line shaded below. (in
  blue)

• Graph each inequality on the same coordinate
  plane. The area where they overlap is the
  solution.

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EXAMPLE 1 CONTINUED

• The region where both shaded areas overlap
  represents the solution to our system. Notice
  the region occurs in both Quadrant II and in
  Quadrant III.
• Any point chosen from this area will produce true
  statements when substituted into both
  inequalities.

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Example 2

• Solve the system by graphing
• The first is an absolute value function use a
  solid line and shade above. (in red)
• The second is a horizontal line use a dashed
  line and shade below. (in blue)
• Since the shaded regions dont overlap, this
  system has no solution.

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Example 3

• Solve the system by graphing
• The first is a vertical line. Use a solid line
  and shade to the right.
• The second is a vertical line. Use a solid line
  and shade to the left.
• The third is a diagonal line. Solve for y. Then
  use a solid line and shade below.
• The solution region is shaded the darkest.

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Up next

• In Lessons 4 and 5, you will study a real-world
  application of solving systems of linear
  equations and inequalities!