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Precalculus

Systems of Equations and Inequalities

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1
Systems of Linear Equations and Inequalities in Two Variables

Cathy can spend at most \(30\) minutes on the treadmill, in some combination of running and walking. To warm up and cool down, she must spend at least \(8\) minutes walking. At the walking speed, she burns off \(3\) calories per minute. At the running speed, she burns off \(8\) calories per minute. Set up the problem that must be solved to answer the following question: How many minutes should Cathy spend on each activity to maximize the total number of calories burned?

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2
Systems of Linear Equations in Three Variables

An investment counselor would like to advise her client about three types of investments: a stock based mutual fund, a corporate bond, and a savings bond. The counselor wants to distribute tot total investment amount according to the clientâ€™s risk tolerance. Risk factors for each investment types range from \(1\) to \(5\), with \(1\) being the most risky, and are summarized in Table 1. The client can tolerate an overall risk level of \(3.5\), and wants the amount invested in the corporate bond to equal the amount invested in the savings bond. Set up and solve a system of equations to determine the percentage of the total investment that should be allocated to each investment type. See Table 1.

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3
Solving Systems of Equations Using Matrices

Perform the indicated row operations (independently of one another, not in succession) on the following augmented matrix.

\(\left[\begin{array}{ccc|c}
0 & 3 & 1 & -1\\
1 & -2 & -1 & 2\\
0 & 6 & -4 & 5
\end{array}\right]\)

Interchange rows \(1\) and \(2\).

Multiply the first row by \(-2\).

Multiply the first row by \(-2\) and add the result to the third row. Retain the original first row.

Find the solution set of the system of linear equations in the variables \(x\), \(y\), \(z\), and \(u\) (in that order) that has the following augmented matrix.

\(\left[\begin{array}{rrrr|r}
1 & 0 & 0 & 3 & 0\\
0 & 1 & 0 & 5 & 1\\
0 & 0 & 1 & -6 & 4
\end{array}\right]\)

The Quilterâ€™s Corner is a company that makes quilted wall hangings, pillows, and bed-spreads. The process of quilting involves cutting, sewing, and finishing. For each of the three products, the numbers of hours spent on each task are given in Table 1 (See book). Every week, there are \(17\) hours available for cutting, \(15\) hours available for sewing, and \(9\) hours available for finishing. How many of each item can be made per week if all the available time is to be used?

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4
Operations on Matrices

Let \(A=\left[\begin{array}{ccc}
-1 & 2 & 0\\
0 & -3.5 & 1
\end{array}\right]\), \(B=\left[\begin{array}{cc}
3 & -2 \\
-1 & 4.2 \\
2 & -6
\end{array}\right]\), \(C=\left[\begin{array}{ccc}
1 & 4 & -5\\
-2.5 & 0 & 2.3
\end{array}\right]\)

Perform the following operations, if defined.

\(A+B\)

\(C-A\)

\(3C\)

Let \(A=\left[\begin{array}{ccc}
1 & -1 & 0\\
2 & -5 & 1
\end{array}\right]\), \(B=\left[\begin{array}{cc}
6 & -2\\
2 & -5\\
-3 & -1\\
1 & -4
\end{array}\right]\), \(C=\left[\begin{array}{cc}
0 & -4\\
2 & 6\\
0 & 1
\end{array}\right]\)

Calculate the following, if defined.

\(AC\)

\(BA\)

\(AB\)

A manufacturer of womenâ€™s clothing makes four different outfits, each of which utilizes some combination of fabrics A, B, and C. The yardage for each outfit is given in matrix \(F\). The cost of each fabric (in dollars per yard) is given in matrix \(C\). Find the total cost of fabric for each outfit.

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5
Matrices and Inverses

If \(A=\left[\begin{array}{ccc}
2 & 0 & -1\\
1 & 4 & 0\\
0 & -2 & 0
\end{array}\right]\), \(B=\left[\begin{array}{ccc}
0 & 1 & 2\\
0 & 0 & -\frac{1}{2}\\
-1 & 2 & 4
\end{array}\right]\),

show that \(AB=I\), where \(I\) is the \(3\times 3\) identity matrix.

Suppose you receive an encoded message in the form of two \(4\times 1\) matrices \(C\) and \(D\).

\(C=\left[\begin{array}{c}
-30\\
109\\
120\\
14
\end{array}\right]\), \(D=\left[\begin{array}{c}
26\\
-81\\
-47\\
-26
\end{array}\right]\)

Find the original message if the encoding was done using the matrix

\(A=\left[\begin{array}{cccc}
-1 & 2 & 0 & -2\\
3 & -7 & 2 & 6\\
2 & -4 & 1 & 7\\
1 & -2 & 0 & 1
\end{array}\right]\)