Solving Systems of Equations Using Substitution

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Lesson 5.2

• Solving Systems of Equations Using Substitution

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• Graphing systems and comparing table values are
  good ways to see solutions.
• It is not always easy to find a good graphing
  window or the right settings for the table to
  view the solution.
• Often these solutions are only approximations.
• To find the exact solutions, youll need to work
  algebraically with the equations.
• Lets look at the substitution method.

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• On a rural highway a police officer sees a
  motorist run a red light at 50 mph and begins
  pursuit. At the instant the police officer
  passes through the intersection at 60 mph, the
  motorist is 0.2 mi down the road. When and where
  will the office catch up to the motorist?
• Write an equation in two variables to model this
  situation.

d the distance from the intersection t time
traveled
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• On a rural highway a police officer sees a
  motorist run a red light at 50 mph and begins
  pursuit. At the instant the police officer
  passes through the intersection at 60 mph, the
  motorist is 0.2 mi down the road. When and where
  will the office catch up to the motorist?
• Solve this system by the substitution method and
  check the solution.

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• When the officer catches up to the motorist they
  will the same distance from the intersection (so
  both equations will have the same d value.

• Replace the d value in one equation with the d
  value from the other equation.

• Solve this new equation for t.

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So at t0.02 hours the police and motorist will
be the same distance from the intersection.
At t0.02 hours both the police and motorist will
be 1.2 miles from the intersection. This is the
only ordered pair that works in both equations.
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All Tied Up

• Materials needed
• Two ropes of different thickness about 1 m long
• Measuring tape
• One 9 meter long thin rope (optional)
• One 10 meter long thick rope (optional)

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Collecting Data and Writing Equations

• Measure the thinner rope without any knots.
• Tie a knot in the rope and measure the length
  again.
• Continue to tie knots and measure the length of
  the rope.
• Record the number of knots and the length of the
  rope in a table.
• Define the variables and write equations in
  intercept form to model the data.
• Repeat for both ropes.

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• Suppose that you had 9 meters of the thinner rope
  and 10 meters of the thicker rope.
• Write a system of equations that gives the length
  of each rope depending on the number of knots
  tied on the rope.

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Analyzing the Data and Solution

• Solve the system of equations using the
  substitution method.
• Select an appropriate window setting and graph
  this system of equations.
• Estimate the coordinates for the point of
  intersection to check your solution.
• Explain the real-world meaning of the solution of
  the system of equations.
• What happens to the graph of the system of the
  two ropes
• Have the same thickness?
• Have the same length?

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• So far you have seen equations written in
  intercept form.
• These equations make it easy to use the
  substitution method since they are already both
  solved for y.
• y900 -6x
• y1000-10.3x
• Sometimes it is necessary to place equations in
  intercept form before using substitution.

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A Mixture Problem

• A pharmacist has 5 saline (salt) solution and
  20 saline solution. How much of each solution
  should be combined to create a bottle of 90 ml of
  10 solution.
• Write a system of equations that models this
  situation.

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• A pharmacist has 5 saline (salt) solution and
  20 saline solution. How much of each solution
  should be combined to create a bottle of 90 ml of
  10 solution.
• Solve the one equation for x or y and substitute
  it into the other equation. Find a solution.

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In this Section

• Saw limitations to solving a problem graphically
• Learned how to solve a system of equations using
  substitution.