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- Systems of linear equations
- Elimination by substitution
Substitution and parallel lines
If computations eliminate all the variables and produce a contradiction, the two lines of a system are parallel, and the system is called inconsistent.
Sample set b
Solve the system
Step 1: Solve equation 1 for
Step 2: Substitute the expression
for
into equation 2.
Step 3: Solve the equation obtained in step 2.
Computations have eliminated all the variables and produce a contradiction. These lines are parallel.
This system is inconsistent.
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Practice set b
Slove the system
Substitution produces
or
, a contradiction. These lines are parallel and the system is inconsistent.
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Substitution and coincident lines
The following rule alerts us to the fact that the two lines of a system are coincident.
Substitution and coincident lines
If computations eliminate all the variables and produce an identity, the two lines of a system are coincident and the system is called dependent.
Sample set c
Solve the system
Step 1: Divide equation 1 by 4 and solve for
Step 2: Substitute the expression
for
in equation 2.
Step 3: Solve the equation obtained in step 2.
Computations have eliminated all the variables and produced an identity. These lines are coincident.
This system is dependent.
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Practice set c
Systems in which a coefficient of one of the variables is not 1 or cannot be made to be 1 without introducing fractions are not well suited for the substitution method. The problem in Sample Set D illustrates this “messy” situation.
Sample set d
Solve the system
Step 1: We will solve equation
for
Step 2: Substitute the expression
for
in equation
Step 3: Solve the equation obtained in step 2.
Step 4: Substitute
into the equation obtained in step
We now have
and
Step 5: Substitution will show that these values of
and
check.
Step 6: The solution is
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Practice set d
Exercises
For the following problems, solve the systems by substitution.
Exercises for review
Questions & Answers
Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.
can you send the book attached ?
Ariel
the study of how humans make choices under conditions of scarcity
AI-Robot
U(x,y) = (x×y)1/2
find mu of x for y
U(x,y) = (xÃy)1/2
find mu of x for y
Desalegn
this is the study of how the society manages it's scarce resources
Belonwu
macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.
husaini
difference between firm and industry
what's the difference between a firm and an industry
Abdul
firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅
Abdulraufu
Suppose the demand function that a firm faces shifted from
Qd 120 3P
to
Qd 90 3P
and the supply function has shifted from
QS
20 2P
to
QS
10 2P .
a) Find the effect of this change on price and quantity.
b) Which of the changes in demand and supply is higher?
explain standard reason why economic is a science
factors influencing supply
scares
means__________________ends
resources. unlimited
Jan
economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses
Jan
calculate the profit maximizing for demand and supply
Why qualify 28 supplies
Milan
out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials
AI-Robot
concepts of supply in microeconomics
identify a demand and a supply curve
there's a difference
Aryan
Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods
Israr
Hi Sir please how do u calculate Cross elastic demand and income elastic demand?
Abari
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Source:
OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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