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7.1 Systems of Equations
Solving by Graphing
Solving Systems of Equations by Graphing
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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
Concept:
A System of Linear Equations
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Any pair of Linear Equations can be a System
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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
A “Break Even Point” Example
A $50 skateboard costs $12.50 to build,
once $15,000 is spent to set up the factory:
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Let x = the number of skateboards
f(x) = 15000 + 12.5x (total cost equation)
g(x) = 50x
(total revenue equation)
Using Algebra to
Check a Proposed Solution
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Is (3,0) also a solution?
Estimating a Solution using
The Graphing Method
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Graph both equations on the same graph paper
If the lines do not intersect, there is no solution
If they intersect:
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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
Approximation …
Solving Systems Graphically
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Practice – Solving by Graphing
Consistent: (1,2)
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y – x = 1  (0,1) and (-1,0)
y + x = 3  (0,3) and (3,0)
(1,2)
Solution is probably (1,2) …
Check it:
2 – 1 = 1 true
2 + 1 = 3 true
therefore, (1,2) is the solution
Practice – Solving by Graphing
Inconsistent: no solutions
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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 it’s an inconsistent system
Practice – Solving by Graphing
Consistent: infinite sol’s
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3y – 2x = 6  (0,2) and (-3,0)
-12y + 8x = -24  (0,2) and (-3,0)
(1,2)
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
The Downside of Solving by Graphing:
It is not Precise
Summary
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Solve Systems by Graphing Them Together
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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
Practice Problems
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Page 372
Problems 15-40, omit 34, 36, & 37