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Systems of Linear Equations (10 Examples) : Equalities, Inequalities and Optimizations

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Find the solution, if it exists, to this system of linear equations:

x + 2y - z = -4
3x + 7y - 6z = -21
x + 4y - 6z = -17

Find the solution, if it exists, to this system of linear equations:

x - z = 2
2x - y = 4
x + y + z = 6

A cookie company makes three kinds of cookies, oatmeal raisin, chocolate chip, and shortbread, packaged in small, medium and large boxes. The small box contains 1 dozen oatmeal raisin and 1 dozen chocolate chip; the medium box has 2 dozen oatmeal raisin, 1 dozen chocolate chip, and 1 dozen shortbread; the large box contains 2 dozen oatmeal raisin, 2 dozen chocolate chip, and 3 dozen shortbread. If you require exactly 15 dozen oatmeal raisin, 10 dozen chocolate chip, and 11 dozen shortbread cookies, how many of each size box should you buy?

Determine which region a, b, c or d represents the graph of the given system of linear inequalities.

6x - 5y ≤ 5
6x + 6y ≤ 60

Graph the system of linear inequalities. Locate the corner points and tell whether the graph is bounded or unbounded.

2x + y ≥ 4
3x + 2y ≥ 6
x ≥ 0; y ≥ 0

Minimize the objective function

C = 2x1 + x2 + 3x3 + x4

Subject to the constraints

x1 + x2 + x3 + x4 ≥ 50
3x1 + x2 + 2x3 + x4 ≥ 100

X1 ≥ 0; X2 ≥ 0; X3 ≥ 0; X4 ≥ 0

Mixture: A brewery manufactures three types of beer - lite, regular and dark. Each vat of lite beer requires 6 bags of barley, 1 bag of sugar, and 1 bag of hops. Each vat of regular beer requires 4 bags of barley, 3 bags of sugar, and 1 bag of hops. Each vat of dark beer requires 2 bags of barley, 2 bags of sugar, and 4 bags of hops. Each day the brewery has 800 bags of barley, 600 bags of sugar, and 300 bags of hops available. The brewery realizes a profit of $10 per vat of lite beer, $20 per vat of regular beer, and $30 per vat of dark beer. How many vats of lite, regular and dark beer should be brewed in order to maximize profits? What is the maximum profit?

Scheduling: An automobile manufacturer must fill orders from two dealers. The first dealer, D1, has ordered 40 cars, while the second dealer, D2, has ordered 25 cars. The manufacturer has the cars stored in two locations, W1 and W2. There are 30 cars in W1 and 50 cars in W2. The shipping costs per car are as follows: $180 from W1 to D1; $150 from W1 to D2; $160 from W2 to D1; $170 from W2 to D2. Under these conditions, how many cars should be shipped from each storage location to each dealer so as to minimize the total shipping costs? What is the minimum shipping cost?

Paying off School Bonds. A school board issues bonds in the amount of $20,000,000 to be retired in 25 years. How much must be paid into a sinking fund at 6% compounded annually to pay off the total amount due?

$6000 is borrowed at 10% compounded semiannually. The amount is to be paid back in 5 years. If a sinking fund is established to repay the loan and interest in 5 years, and the fund earns 8% compounded quarterly, how much will have to be paid into the fund every 3 months?

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Solution Summary

Systems of linear equations represented by equalities, inequalities and optimizations are presented in detail. Graph diagrams are included. The solution is detailed and well presented.

Solution provided by:
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
Recent Feedback
  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
  • "excellent work"
  • "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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