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Using Systems of Linear Equations to Solve Word Problems

Hello, I need help understanding how to apply linear systems to the following word problems:

1. The M-Disco company learns that each van now costs 13,000 and each small truck $18,000. The company decides to buy only 182 new Vehicles. How many of each kind should it buy?

2. Matt took his clothes to the cleaners three times last month. First he brought 3 shirts and 1 pair of slacks and paid $10.96. Then he bought 7 shirts, 2 pairs of slacks and a sports coat and paid $30.40. Finally, he brought 4 shirts and 1 sports coat and paid $14.45. How much was he charged for each shirt, each pair of slacks and each sports coat?

3. An investor wants to invest 30,000 in corporate bonds that are rated AAA, A and B . The lower rated ones pay higher interest, but pose a higher risk as well. The average yield is 5% on AAA bonds ,6% on A bonds and 10% on B bonds. Being conservative, the investor wants to have twice as much in AAA bonds as in B bonds .How much should she invest in each type of bond to have an interest income of $2000?

4. Chicken pies cost $3 per pound, dried fruit $4 per pound and nuts $8 per pound . How many pounds of each should be used to produce 140 pounds of trail mix costing $6 per pound in which there are twice as many pretzels by weight as dried fruit?

5. In November of 2008, HBO released the complete series of 'The Sopranos' on DVD. According to an ad in the New York times there were 86 episodes on 33 discs. If 5 discs were sound tracks and bonus materials and 6 discs each had 4 episodes, how many discs had 2 episodes and how many had 3?

Solution Preview

1. Solution:
Let x be the number of vans and y be the number of small trucks.
There is some information missing regarding the total price of all cars and small trucks in question. Please send complete question. I will update posting then.

2. Solution:
Let x be the price of shirt, y be the price of pair of slacks and z be the price of each sports coat.
3 shirts and 1 pair of slacks and paid $10.96.
Equation will be 3x + y = 10.96
7 shirts ,2 pairs of slacks and a sports coat and paid $30.40
Equation will be 7x + 2y + z = 30.40
4 shirts and 1 sports coat and paid $14.45.
Equation will be 4x + z = 14.45
System of equations will be
3x + y = 10.96-------------(i)
7x + 2y + z = 30.40------(ii)
4x + z = 14.45----------(iii)
Subtract equation (iii) from (ii), we will get
(7x+2y+z)-(4x+z) = 30.40 -14.45
3x+2y = 15.95------------(iv)
Now consider equations
3x + y = 10.96-------------(i)
3x+2y = 15.95------------(iv)
Subtract (i) from (iv), we will get
y = $4.99
Now from (i)
3x + y = 10.96
3x + 4.99 = 10.96
3x = 5.97
x = 5.97/3 = $1.99
Now from (iii)
4x + z = 14.45
7.96 ...

Solution Summary

The expert examines using systems of linear equations to solve word problems.