7.1 Systems of Equations

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7.1 Systems of Equations

• Solving by Graphing

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Solving Systems of Equations by Graphing

• What is a System of Equations?
• Solving Linear Systems The Graphing Method
• Consistent Systems one point (x,y) solution
• Inconsistent Systems no solution
• Dependant Systems infinite solutions
• Solving Equations Graphically

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ConceptA System of Linear Equations

• Any pair of Linear Equations can be a System
• A Solution Point is an ordered pair (x,y) whose
  values make both equations true
• When plotted on the same graph, the solution is
  the point where the lines cross (intersection)
• Some systems do not have a solution

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Why Study Systems of Equations?
We will study systems of 2 equations in 2
unknowns (usually x and y) The algebraic methods
we use to solve them will also be useful in
higher degree systems that involve quadratic
equations or systems of 3 equations in 3 unknowns
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A Break Even Point Example A 50 skateboard
costs 12.50 to build, once 15,000 is spent to
set up the factory

• Let x the number of skateboards
• f(x) 15000 12.5x (total cost equation)
• g(x) 50x (total revenue
  equation)

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Using Algebra toCheck a Proposed Solution
Is (3,0) also a solution?
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Estimating a Solution usingThe Graphing Method

• Graph both equations on the same graph paper
• If the lines do not intersect, there is no
  solution
• If they intersect
• Estimate the coordinates of the intersection
  point
• Substitute the x and y values from the (x,y)
  point into both original equations to see if they
  remain true equations

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Approximation Solving Systems Graphically
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Practice Solving by Graphing
Consistent (1,2)
y x 1 ? (0,1) and (-1,0) y x 3 ?
(0,3) and (3,0) Solution is probably (1,2)
Check it 2 1 1 true 2 1 3
true therefore, (1,2) is the solution
(1,2)
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Practice Solving by Graphing
Inconsistent no solutions
y -3x 5 ? (0,5) and (3,-4) y -3x 2 ?
(0,-2) and (-2,4) They look parallel No
solution Check it m1 m2 -3 Slopes are
equal therefore its an inconsistent system
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Practice Solving by Graphing
Consistent infinite sols
3y 2x 6 ? (0,2) and (-3,0) -12y 8x -24
? (0,2) and (-3,0) Looks like a dependant system
Check it divide all terms in the 2nd
equation by -4 and it becomes identical to the
1st equation therefore, consistent, dependant
system
(1,2)
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The Downside of Solving by Graphing It is not
Precise
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Summary

• Solve Systems by Graphing Them Together
• Graph neatly both lines using x y intercepts
• Solution Point of Intersection (2 Straight
  Lines)
• Check by substituting the solution into all
  equations
• Cost and Revenue lines cross at Break Even
  Point
• A Consistent System has one solution (x,y)
• An Inconsistent System has no solution The lines
  are Parallel (have same slope, different
  y-intercept)
• A Dependent System happens when both equations
  have the same graph (the lines have same slope
  and y-intercept)
• Graphing can solve equations having one variable

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Practice Problems

• Page 372
• Problems 15-40, omit 34, 36, 37