1. Solve the following system of equations by elimination.
Answer: x = .5; y = 1.67
Solution:
Rewrite in order to align the x and y terms
Add the second equation to the first equation and solve for x.
Substitute the value obtained for x into either of the original equations.
or
2. Solve the following system of equations by elimination.
Answer: x = -2; y = 5
Solution:
Multiply the first equation by 2.
Subtract the first equation from the second equation and solve for y.
Substitute the value obtained for y into either of the original equations.
or
3. Solve the following system of equations by elimination.
1st
original equation
2nd
original equation
3rd original equation
Answer: x1 = 2; x2 = 1; x3 = -2
Solution:
Part A First eliminate x3.
Step 1. Add the 1st original equation and the 3rd original equation.
+
=
Step 2. Multiply the 2nd original equation by 2, multiply the 3rd original equation by 5 and add the 2nd original equation to the 3rd original equation.
+
=
Part B. From Part A there are two new equations.
1st new equation
2nd
new equation
Step 1. Multiply the 1st new equation by 7, subtract the second new equation from the first new equation, and solve for x2.
_
=
Part C. Solve for x1 by substituting the value received for x2 in either of the new equations in Part B.
1st
new equation
or
2nd
new equation
Part D. Solve for x3 by substituting the values obtained for x1 and x2 in any of the original equations.
1st
original equation
or
2nd
original equation
or
3rd
original equation
4. A manufacturer produces two items which are sold at prices of p1 dollars and p2 dollars each. The supply and demand functions in units for the items are as follows:
P(s1) = the supply function for the first item.
P(d1) = the demand function for the first item.
P(s2) = the supply function for the second item.
P(d2) = the demand function for the second item.
How should prices be set for each item to equate supply and demand? What are the equilibrium quantities for each item?
Answer: The price of the first item should be set at $1.80 and the price of the second item should be set at $1.50. The equilibrium quantity for the first item is 82 units and the equilibrium quantity for the second item is 181 units.
Solution:
Part A.
Step 1. Set the supply function for item 1 equal to the demand function for item 1 and collect terms.
P(s1) = P(d1)
Step 2. Set the supply function for item 2 equal to the demand function for item 2 and collect terms.
P(s2) = P(d2)
Part B. From Part A the following system of equations has been obtained.
Solve this system for the prices by elimination.
1st equation from Part A
2nd equation from part A
Step 1. Multiply the 1st equation by 7 ,add the 1st and second equations, and solve for p1.
+
=
Step 2 . Substitute the value obtained for p1 in either equation from Part A and solve for p2.
1st
equation from Part A
or
2nd equation from part A
Part C. Determine equilibrium for each item by substituting the prices obtained in Part B in either the demand function or the supply function.
For item 1
or
For item 2
or
[Index]