Transcript Section 1.7
CHAPTER 1: FUNCTIONS,
GRAPHS, AND MODELS; LINEAR
FUNCTIONS
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Section 1.7: Systems of Linear Equations in Two
Variables
SECTION 1.7: Systems of Linear
Equations in Two Variables
The break even point is a term used to represent
the point at which the Revenue = Cost. (R(x) =
C(x))
If we know R(x) and C(x), we can find the point at which
the functions ‘meet’ graphically.
Graph both functions and identify the point of
intersection – this marks the break even point.
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SECTION 1.7: Systems of Linear
Equations in Two Variables
Suppose a company has its total revenue, in
dollars, for a product given by
R(x) = 5585x
and its total cost in dollars is given by
C(x) = 61,740 + 440x
where x is the number of thousands of tons of the
product that is produced and sold each year.
Determine the break even point and the corresponding
values for R and C. Interpret.
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SECTION 1.7: Systems of Linear
Equations in Two Variables
Let Y1 = R(x) and Y2 = C(x).
Graph both functions – be sure to use a window
that allows you to locate the intersection.
Find the point of intersection.
2nd TRACE (CALC)
5: intersect
Follow First and Second Curve Prompts (use the up and
down arrows to move from curve to curve)
Move the cursor to an approximate location for the guess.
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SECTION 1.7: Systems of Linear
Equations in Two Variables
Will two lines always intersect?
If 2 equations graph the same line, we say the system is a
dependent system.
If 2 equations graph parallel lines, we say the system is
inconsistent.
There are an infinite number of solutions for such as system.
There is No solution for such a system.
If 2 equations graph with an intersection, we say the system is
consistent.
There is one unique solution for such a system.
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SECTION 1.7: Systems of Linear
Equations in Two Variables
Solution by Substitution
Solve one of the equations for one of the variables in terms of
the other variable.
Substitute the expression from step 1 into the other equation
to give an equation in one variable.
Solve the linear equation for the variable.
Substitute this solution into the equation from step 1 or into
one of the original equations and solve this equation for the
second variable.
Check the solution in both original equations or check
graphically.
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SECTION 1.7: Systems of Linear
Equations in Two Variables
Solve the system below by substitution:
3x 4y 10
4x 2y 6
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SECTION 1.7: Systems of Linear
Equations in Two Variables
Solution by Elimination
If necessary, multiply one or both equations by a nonzero
number that will make the coefficients of one of the variables
in the equations equal, except perhaps for the sign.
Add or subtract the equations to eliminate one of the
variables.
Solve for the variable in the resulting equation.
Substitute the solution from step 3 into one of the original
equations and solve for the second variable.
Check the solutions in the remaining original equation, or
graphically.
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SECTION 1.7: Systems of Linear
Equations in Two Variables
Solve the system below by elimination:
3x 4y 10
4x 2y 6
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SECTION 1.7: Systems of Linear
Equations in Two Variables
Market equilibrium
Demand is the quantity of a product demanded by
consumers.
Supply is the quantity of a product supplied .
Both demand and supply are related to the price.
Equilibrium price is the price at which the number of units
demanded equals the number of units supplied.
We can also refer to this as market equilibrium.
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SECTION 1.7: Systems of Linear
Equations in Two Variables
Suppose the daily demand for a product is given by
p = 200 – 2q, where q is the number of units
demanded and p is the price per unit dollars. The
daily supply is given by p = 60 + 5q, where q is the
number of units supplied and p is the price in
dollars.
If the price is $140, how many units are supplied and how
many are demanded?
Does this price result in a surplus or shortfall?
What price gives market equilibrium?
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SECTION 1.7: Systems of Linear
Equations in Two Variables
A nurse has two solutions that contain different
concentrations of certain medication. One is a 12%
concentration, and the other is an 8%
concentration. How many cubic centimeters (cc) of
each should she mix together to obtain 20cc of a
9% solution?
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SECTION 1.7: Systems of Linear
Equations in Two Variables
An investor has $300,000 to invest, part at 12% and
the remainder in a less risky investment at 7%. If
her investment goal is to have an annual income of
$27,000, how much should she put in each
investment?
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SECTION 1.7: Systems of Linear
Equations in Two Variables
Inconsistent and Dependent Systems
Use elimination to solve each system, if possible.
2x 3 y 4
6 x 9 y 12
2x 3y 4
6x 9y 36
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SECTION 1.7: Systems of Linear
Equations in Two Variables
Homework: pp. 110-114
1-29 every other odd, 33, 37, 41, 45, 49, 53, 57
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