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

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?