Ms. Sommerman

1
Solving Linear Systems Using Graphing

• Ms. Sommerman
• Algebra 1
• 2005-2006

2
Solving Systems of Linear Equations
TAKE THE QUIZ!!
This presentation is designed to help you review
the material we covered in class. Choose where
you would like to begin. Take the quiz when you
are ready! You will be handing in your work at
the end of the presentation.
3
What is a linear system?

• A system of linear equations is two or more
  linear equations that are being solved
  simultaneously. 
• The simplest linear system to work with is in two
  variables (x, y).
• Example 2x 4y 3
• 7x 10y 2

4
What is a solution of linear system?

• A solution of a system in two variables is an
  ordered pair (x,y) that makes BOTH equations
  true. 
• If an ordered pair is a solution to one equation,
  but not the other, then it is NOT a solution to
  the system.

5
How do you determine if an ordered pair is a
solution to a system?

• STEPS
• Substitute the ordered pair in for (x, y) into
  the first linear equation. See if it makes the
  equation true.
• 2. Substitute the same ordered pair in for (x,
  y) in the second linear equation. See if it
  makes that equation true.
• 3. If the ordered pair makes both equations
  true, it is a solution.

6
Example 1 Solutions of Linear Systems

• Determine whether the ordered pair (3,-1) is a
  solution of the system xy 2 (Eq. 1)
• x-y 4 (Eq. 2)
• Substitute the ordered pair into both linear
  equations for x and y.
• (Eq 1) 3 (-1) 2 (Eq 2) 3 (-1) 4
• Solve the equations and determine if the result
  is a true or false statement.
• (Eq. 1) 2 2 (TRUE) (Eq. 2) 4 4 (TRUE)
• Because the ordered pair made both equations
  true, IT IS A SOLUTION!

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Example 2 Solutions of Linear Systems

• Determine whether the ordered pair (2,-1) is
  a solution of the system 3x4y 2 (Eq. 1)
• x-8y 4 (Eq. 2)
• Substitute the ordered pair into both linear
  equations for x and y.
• (Eq. 1) 3( 2) 4(-1) 2 (Eq. 2) 2 8(-1) 4
• Solve the equations and determine if the result
  is a true or false statement.
• (Eq. 1) 6 (-4) 2 (Eq. 2) 2 8 4
• 2 2 (TRUE) 104 (FALSE, 10
• DOES NOT EQUAL 4)
• Because the ordered pair does not make both
  equations true, IT IS NOT A SOLUTION!

8
Your Turn!! Try this one on your own.

• Determine whether the ordered pair (4,5) is a
  solution of the system 3x4y 2
• x-8y 4
• Click on the correct answer.
• IT MOST DEFINITELY IS!
• IT MOST DEFINITELY IS NOT!

9
INCORRECT!

• When you substitute the ordered pair in for x and
  y in both equations. It should make the
  equations true. If it does not, it is not a
  solution.

10
GREAT JOB!!!Try another one on your own.

• Determine whether the ordered pair (3,-1) is a
  solution of
• the system y x 3
• y -x 7
• Click on the correct answer.
• OF COURSE! NOT IN A MILLION
  YEARS!

11
What is a solution of linear system?

• A solution of a system in two variables is an
  ordered pair (x,y) that makes BOTH equations
  true. 
• If an ordered pair is a solution to one equation,
  but not the other, then it is NOT a solution to
  the system.

12
IM SO SAD!! Thats incorrect!

• Please review the definition of a solution.

13
ROCK ON! Thats correct!

Go back to the homepage and decide where you want
to go next!
14
Solving Linear Systems by Graphing

• Because the solution of a linear system satisfies
  each equation in the system, the solution must
  lie on the graphs of both equations.
• The solution to the system is point where the two
  lines cross.
• (-2,-2) is the solution to the system to the
  right. When (-2,-2) is substituted into
  either equation, it will make them both true.

15
Solving a Linear System Using Graphing

• STEPS
• Write each equation in y mx b.
  Determine the slope and y-intercept of each
  equation
• Graph both equations on the same coordinate
  plane.
• Determine the coordinates of the point of
  intersection.
• Check the coordinates algebraically by
  substituting into each equation of the original
  linear system.

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Example 1 Graphing Linear Systems

• Solve Graphically 4x-6y12
  2x2y6
• Write each equation in y mx b. Determine the
  slope and y-intercept of each equation.

17
Example 1 Graphing Linear Systems Continued

• Solve Graphically 4x-6y7
  2x2y1
• 2. Graph each equation on the
• same coordinate plane.
• 3. Determine the point of
• Intersection.
• The point of intersection of the
• two lines (1,-1/2) is the solution to
• the system of equations.
• 4. Check the coordinates
• algebraically by substituting into
• each equation of the original
• linear system.
• When (1,-1/2) is substituted into
• either equation, it made them
• both true.

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Example 2 Graphing Linear Systems

• Solve Graphically y  x
• 3y 4x 2
• Write each equation in y mx b. Determine the
  slope and y-intercept of each equation

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Example 2 Graphing Linear Systems Continued

• Solve Graphically y  x
• 3y 4x 2
• 2. Graph each equation on the
• same coordinate plane.
• 3. Determine the point of
• Intersection.
• The point of intersection of the
• two lines (-2,-2) is the solution to
• the system of equations.
• 4. Check the coordinates
• algebraically by substituting into
• each equation of the original
• linear system.
• When (-2,-2) is substituted into
• either equation, it made them
• both true.

20
Your Turn!! You will need paper and pencil to
graph the following problems.  If you have graph
paper, please use your graph paper.  Do NOT use
your graphing calculator for these practice
problems. Try this one on your own.

• Solve Graphically 2x - 2y 1
• 3y 3x 7
• Click on the correct answer.

21
TRY AGAIN!Thats incorrect!

• Please review how to graph a linear system.

22
Solving a Linear System Using Graphing

• STEPS
• Write each equation in y mx b.
  Determine the slope and y-intercept of each
  equation
• Graph both equations on the same coordinate
  plane.
• Determine the coordinates of the point of
  intersection.
• Check the coordinates algebraically by
  substituting into each equation of the original
  linear system.

23
Fantabolous!Try another one!

• Solve Graphically 8x - y 1/5
• y -4/5x 13/10
• Click on the correct answer.

24
IM ALL SMILES!! Thats correct!

Go back to the homepage and decide where you want
to go next!
25
INCORRECT!

• Check your work by checking the solution
  algebraically. When you substitute the ordered
  pair in for x and y in both equations it should
  make the equations true. If it does not it is
  not a solution.

26
Quiz Time!!!

You will need paper, pencil, and graph paper to
answer the following questions.  If you have
graph paper, USE IT!  Do NOT use your graphing
calculator! You will be handing in you work at
the end of the quiz. Good luck!
27
HERE WE GO!!

• On a graph, the solution to a system of linear
  equations will be represented by ______.
• A. the xintercept of each line
• B. the yintercept of each line
• C. all the points on either line
• D. the intersection of the lines

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ID ERASE THAT!

• Think about where a solution is located on a
  graph of two linear equations.
• TRY AGAIN!

29
YOURE IN THE MONEY!! THATS RIGHT!

• Explanation On a graph, the solution to a system
  of linear equations will be represented by the
  intersection of the lines. The intersection of
  the lines indicates which point or points satisfy
  each of the equations. Go on to the next question.

30
INCORRECT!

• Check your work by checking the solution
  algebraically. When you substitute the ordered
  pair in for x and y in both equations. It should
  make the equations true. If it does not it is
  not a solution.

31
Try this one!
2. The point (2, 1) is a solution to which of
the following systems of
equations? A. 2x y 5 and -2x y 2 B.
-2x - y -5 and 2x - y 2 C. -x y 3 and
-3x y -5 D. x y 3 and 3x - y 5
32
YAH-HOO! You answered this question correctly!!

• Explanation The point (2, 1) is a solution to
• x y 3 and 3x - y 5. If you substitute the
  coordinates into each system, it will make both
  equations true. Go on to the next one!

33
GRANNY IS VERY DISAPPOINTED! Thats incorrect!

• Please review how to determine if an ordered pair
  is a solution to a system.

34
How do you determine if an ordered pair is a
solution to a system?

• STEPS
• Substitute the ordered pair in for (x, y) into
  the first linear equation. See if it makes the
  equation true.
• 2. Substitute the same ordered pair in for (x,
  y) in the second linear equation. See if it
  makes that equation true.
• 3. If the ordered pair makes both equations
  true, it is a solution.

35
You think youre so smart. Try this one!

• Decide whether (-3, 5) is a solution of the
  linear system 2x- 5y -31
• -3x y 14
• Yes it is!
• No, its not!

36
ARE YOU KIDDING! Thats incorrect!

• Please review how to determine if an ordered pair
  is a solution to a system.

37
How do you determine if an ordered pair is a
solution to a system?

• STEPS
• Substitute the ordered pair in for (x, y) into
  the first linear equation. See if it makes the
  equation true.
• 2. Substitue the same ordered pair in for (x, y)
  in the second linear equation. See if it makes
  that equation true.
• 3. If the ordered pair makes both equations
  true, it is a solution.

38
Way to go! You need graph paper for this one.
4. Solve Graphically 5x 8y 1 3x
5y 1 Click on the correct answer.
39
YOU MADE THE BABY CRY! Thats incorrect!

• Please review
• how to graph a
• linear system.

40
Solving a Linear System Using Graphing

• STEPS
• Write each equation in y mx b.
  Determine the slope and y-intercept of each
  equation
• Graph both equations on the same coordinate
  plane.
• Determine the coordinates of the point of
  intersection.
• Check the coordinates algebraically by
  substituting into each equation of the original
  linear system.

41
GREAT JOB! Try this one.

• 5. Solve Graphically -2x 8y -2
• -x 5y 1
• Click on the correct answer.

42
ARE YOU AWAKE! Thats incorrect!

• Please review how to solve a linear system using
  graphing.

43
Solving a Linear System Using Graphing

• STEPS
• Write each equation in y mx b.
  Determine the slope and y-intercept of each
  equation
• Graph both equations on the same coordinate
  plane.
• Determine the coordinates of the point of
  intersection.
• Check the coordinates algebraically by
  substituting into each equation of the original
  linear system.

44
YOURE TOO SMART!! THATS RIGHT!Try another
one!

• 6. Solve using substitution x - 3y -2
• 2x 2y 2
• Click on the correct answer.
• A. (1, 3)
• B. (2.5, 1.5)
• C. (4.5, 2)

45
Im going to send Mr. T after you if you dont
get this right!

• Please review how to solve a linear system using
  substitution linear system.

46
Solving a Linear System Using Substitution

• STEPS
• Solve one of the equations for one of its
  variables.
• Substitute the expression from Step 1 into the
  equation you did not use.
• Solve for the other variable.
• Substitute the value from Step 3 into the
  revised equation from Step 1 and solve.
• Check the solution in each of the original
  equations.

47
Thats one happy frog! Your right! Try another
one!

• 7. Solve using substitution -2x - 3y -2
• -2x 4y 6
• Click on the correct answer.
• A. (-1, 3)
• B. (13, -8)
• C. (-13, 8)

48
Hes not too happy anymore! Thats incorrect!

• Please review how to solve a linear system using
  substitution.

49
Solving a Linear System Using Substitution

• STEPS
• Solve one of the equations for one of its
  variables.
• Substitute the expression from Step 1 into the
  equation you did not use.
• Solve for the other variable.
• Substitute the value from Step 3 into the
  revised equation from Step 1 and solve.
• Check the solution in each of the original
  equations.

50
Hes soo happy! Thats correct!Try another one!

• 8. Solve using elimination 3x 2y 8
• 2y 12 5x
• Click on the correct answer.
• A. (2, -1)
• B. (-2, -1)
• C. (2, 1)

51
Solving a Linear System Using Elimination

• STEPS
• 1. Rearrange each equation
• so the variables are on one side
• (in the same order) and the
• constant is on the other side.
• 2. Multiply one or both equations
• by an integer so that one term
• has equal and opposite
• coefficients in the two equations.
• 3. Add the equations to
• produce a single equation
• with one variable.

• 4. Solve for the variable.
• 5. Substitute the variable back
• into one of the equations and
• solve for the other variable.
• 6. Check the solution--it should
• satisfy both equations.

52
Hes sad now! Thats incorrect!

• Please review how to solve a linear system using
  elimination.

53
CORRECT! He wants to give you a big kiss for
that one! Try another one!

• 9. Solve using elimination 3x 5y 6
• -4x 2y 5
• Click on the correct answer.
• A. (-.5, -1.5)
• B. (-.5, 1.5)
• C. (1, 1)

54
Solving a Linear System Using Elimination

• STEPS
• 1. Rearrange each equation
• so the variables are on one side
• (in the same order) and the
• constant is on the other side.
• 2. Multiply one or both equations
• by an integer so that one term
• has equal and opposite
• coefficients in the two equations.
• 3. Add the equations to
• produce a single equation
• with one variable.

• 4. Solve for the variable.
• 5. Substitute the variable back
• into one of the equations and
• solve for the other variable.
• 6. Check the solution--it should
• satisfy both equations.

55
I dont think hes very happy! Thats incorrect!

• Please review how to solve a linear system using
  elimination.

56
CONGRATULATIONS!!ALL DONE!!!!

• You have successfully completed the quiz! I hope
  you feel ready for test!

57
Solving Linear Systems by Substitution

• Graphing is a useful tool for solving systems of
  equations, but it can sometimes be
  time-consuming.
• A quicker way to solve systems is to isolate one
  variable in one equation, and substitute the
  resulting expression for that variable in the
  other equation.

58
Solving a Linear System Using Substitution

• STEPS
• Solve one of the equations for one of its
  variables.
• Substitute the expression from Step 1 into the
  equation you did not use.
• Solve for the other variable.
• Substitute the value from Step 3 into the
  revised equation from Step 1 and solve.
• Check the solution in each of the original
  equations.

59
Example 1 Solving Linear Systems Using
Substitution

• Solve using substitution x y 1
  -2x y 2

60
Example 2 Solving Linear Systems Using
Substitution
Solve using substitution y 4
-2x y 2
61
Your Turn!! You will need paper and pencil to
solve the following problems.  Try this one on
your own.

• Solve using substitution -4x - 5y 3
• x y 2
• Click on the correct answer.
• A. (13, -11)
• B. (-13, 11)
• C. (12, 10)

62
THATS PAINFUL and incorrect!

• Please review how to solve a linear system by
  substitution.

63
Solving a Linear System Using Substitution

• STEPS
• Solve one of the equations for one of its
  variables.
• Substitute the expression from Step 1 into the
  equation you did not use.
• Solve for the other variable.
• Substitute the value from Step 3 into the
  revised equation from Step 1 and solve.
• Check the solution in each of the original
  equations.

64
YOURE TOO SMART!! THATS RIGHT!Try another
one!

• Solve using substitution x - 3y -2
• 2x 2y 2
• Click on the correct answer.
• A. (1, 3)
• B. (2.5, 1.5)
• C. (4.5, 2)

65
THEYRE HAPPY because thats correct!

Go back to the homepage and decide where you want
to go next!
66
THATS SCARY and INCORRECT!

• Check your work by checking the solution
  algebraically. When you substitute the ordered
  pair in for x and y in both equations it should
  make the equations true. If it does not it is
  not a solution.

67
Solving Linear Systems by Elimination

• In the elimination method, the two equations in
  the system are added to create a new equation
  with only one variable.
• In order for the new equation to have only one
  variable, the other variable must cancel out.
• In other words, we must first perform operations
  on each equation until one term has an equal and
  opposite coefficient as the corresponding term in
  the other equation.

68
Solving a Linear System Using Elimination

• STEPS
• 1. Rearrange each equation
• so the variables are on one side
• (in the same order) and the
• constant is on the other side.
• 2. Multiply one or both equations
• by an integer so that one term
• has equal and opposite
• coefficients in the two equations.
• 3. Add the equations to
• produce a single equation
• with one variable.

• 4. Solve for the variable.
• 5. Substitute the variable back
• into one of the equations and
• solve for the other variable.
• 6. Check the solution--it should
• satisfy both equations.

69
Example 1 Solving Linear Systems Using
Elimination

• Solve using elimination 2y - 3x 7  and 5x
  4y - 12

• Rearrange each equation-3x 2y 75x - 4y
  - 12
• Multiply the first equation
• by 2-6x 4y 145x - 4y - 12
• Add the equations- x 2
• Solve for the variablex - 2

• Plug x - 2 into one of the equations and solve
  for y
• -3(- 2) 2y 7
• 6 2y 7
• 2y 1
• y 1/2  
• Thus, the solution to the
• system of equations is (- 2, 2 ).
• 6. Check2( 1/2 ) - 3(- 2) 7 ? Yes.5(- 2)
  4( 1/2) - 12 ? Yes.

70
Example 2 Solving Linear Systems Using
Elimination

• Solve using elimination 4y - 5 20 - 3x and
  4x - 7y 16 0

1. Rearrange each equation 3x 4y 25 4x -
7y - 16 2. Multiply the first equation by 4
and the second equation by -3 12x 16y
100 -12x 21y 48 3. Add the equations37y
148 4. Solve for the variabley 4

• 5. Plug y 4 into one of the equations and
  solve for x
• 3x 4(4) 25
• 3x 16 25
• 3x 9
• x 3
• Thus, the solution to the system of
• equations is (3, 4).
• 6. Check
• 4(4) - 5 20 - 3(3) ? Yes.
• 4(3) - 7(4) - 16 ? Yes.

71
Your Turn!! You will need paper and pencil to
solve the following problems.  Try this one on
your own.

• Solve using elimination 2x 4y36 
• 10y - 5x0
• Click on the correct answer.
• A. (9, -4.5)
• B. (9, 4.5)
• C. (1, 7)

72
Hmm.I dont think thats correct.

• Please review how to solve a linear system by
  elimination.

73
Solving a Linear System Using Elimination

• STEPS
• 1. Rearrange each equation
• so the variables are on one side
• (in the same order) and the
• constant is on the other side.
• 2. Multiply one or both equations
• by an integer so that one term
• has equal and opposite
• coefficients in the two equations.
• 3. Add the equations to
• produce a single equation
• with one variable.

• 4. Solve for the variable.
• 5. Substitute the variable back
• into one of the equations and
• solve for the other variable.
• 6. Check the solution--it should
• satisfy both equations.

74
Hes a little too excited that you got that one
right.Try another one!

• Solve using elimination x - 3y -2
• 2x 2y 2
• Click on the correct answer.
• A. (1, 3)
• B. (2.5, 1.5)
• C. (4.5, 2)

75
Solving a Linear System Using Elimination

• STEPS
• 1. Rearrange each equation
• so the variables are on one side
• (in the same order) and the
• constant is on the other side.
• 2. Multiply one or both equations
• by an integer so that one term
• has equal and opposite
• coefficients in the two equations.
• 3. Add the equations to
• produce a single equation
• with one variable.

• 4. Solve for the variable.
• 5. Substitute the variable back
• into one of the equations and
• solve for the other variable.
• 6. Check the solution--it should
• satisfy both equations.

76
Hes happy too!

Go back to the homepage and decide where you want
to go next!
77
He is not happy!

• Check your work by checking the solution
  algebraically. When you substitute the ordered
  pair in for x and y in both equations it should
  make the equations true. If it does not it is
  not a solution.