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Precalculus
Systems of Equations and Inequalities

1
Systems of Linear Equations and Inequalities in Two Variables

Problem  1
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Problem  2
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Problem  3
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Graph the following set of each of the following linear inequalities.

  1. \(x \geq 3\)

  2. \(y < 2x\)

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Problem  4
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Problem  5

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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Problem  6
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2
Systems of Linear Equations in Three Variables

Problem  1
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Problem  2
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Problem  3
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Problem  4
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Problem  5

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 client’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

Problem  1

Perform the indicated row operations (independently of one another, not in succession) on the following augmented matrix.
\(\left[\begin{array}{ccc|c} 0 & 3 & 1 & -1\\ 1 & -2 & -1 & 2\\ 0 & 6 & -4 & 5 \end{array}\right]\)

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

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

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

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Problem  2
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Problem  3
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Problem  4
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Problem  5
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Problem  6
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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.
\(\left[\begin{array}{rrrr|r} 1 & 0 & 0 & 3 & 0\\ 0 & 1 & 0 & 5 & 1\\ 0 & 0 & 1 & -6 & 4 \end{array}\right]\)

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Problem  7

The Quilter’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

Problem  1
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Let \(B=\left[\begin{array}{cccc} -2 & 4 & 7 & 0\\ 0 & 5 & 12 & 6\\ -8 & -7 & 0 & 1 \end{array}\right]\)
Find the following:

  1. The dimensions of \(B\)

  2. The value of \(b_{31}\)

  3. The value of \(b_{13}\)

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Problem  2
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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.

  1. \(A+B\)

  2. \(C-A\)

  3. \(3C\)

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Problem  3
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Find the product \(p\) if
\(R=\left[\begin{array}{cccc} 3 & 2 & 3 & -2 \end{array}\right]\) and \(C=\left[\begin{array}{c} 2\\ -2\\ 3\\ 1 \end{array}\right]\)

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Problem  4
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Find the product \(AB\) for the following matrices.
\(A=\left[\begin{array}{cccc} 3 & 4 & 0 & -2\\ -1 & 0 & -3 & 2 \end{array}\right]\), \(B=\left[\begin{array}{cc} 5 & 0\\ -4 & 1\\ 2 & 6\\ 0 & -2 \end{array}\right]\)

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Problem  5
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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.

  1. \(AC\)

  2. \(BA\)

  3. \(AB\)

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Problem  6

A manufacturer of women’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

Problem  1

Let \(A=\left[\begin{array}{cc} -2 & 3\\ 4 & 7 \end{array}\right]\)
Show that \(IA=A\).

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Problem  2
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Find the inverse of \(A=\left[\begin{array}{cc} 1 & -3\\ -2 & 5 \end{array}\right]\)

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Problem  3

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.

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Problem  4
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Find the inverse of \(A=\left[\begin{array}{ccc} 3 & 3 & 9\\ 1 & 0 & 2\\ -2 & 3 & 0 \end{array}\right]\)

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Problem  5
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Problem  6
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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
\(A=\left[\begin{array}{cccc} -1 & 2 & 0 & -2\\ 3 & -7 & 2 & 6\\ 2 & -4 & 1 & 7\\ 1 & -2 & 0 & 1 \end{array}\right]\)

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6
Determinants and Cramer's Rule

Problem  1
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Evaluate \(\left|\begin{array}{cc} -3 & -2\\ 4 & 6 \end{array}\right|\)

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Problem  2
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Let \(A=\left[\begin{array}{ccc} 0 & 1 & 3\\ -2 & 5 & 7\\ 4 & 0 & -1 \end{array}\right]\)

  1. Find \(M_{12}\) and \(C_{12}\)

  2. Find \(M_{13}\) and \(C_{13}\)

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Problem  3
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Evaluate the determinant of \(A\).
\(A=\left[\begin{array}{ccc} 0 & 1 & 3\\ -2 & 5 & 7\\ 4 & 0 & -1 \end{array}\right]\)

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Problem  4
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Problem  5
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7
Partial Fractions

Problem  1
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Compute the partial fraction decomposition of \(\dfrac{12}{x^2-4}\)

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Problem  2
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Write the partial fraction decomposition of \(\dfrac{4x^2-7x+1}{x^3-2x^2+x}\)

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Problem  3
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Check whether the quadratic polynomial \(2x^2+x+3\) is irreducible.

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Problem  4
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Write the partial fraction decomposition of \(\dfrac{3x^2-2x+13}{(x+1)(x^2+5)}\)

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Problem  5
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Find the partial fraction decomposition of \(\dfrac{-2x^3+x^2-6x+7}{(x^2+3)^2}\)

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8
Systems of Nonlinear Equations

Problem  1
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Problem  2
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Problem  3
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Problem  4
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A right triangle with a hypotenuse of \(2\sqrt{15}\) inches has an area of \(15\) square inches. Find the lengths of the other two sides of the triangle. See Figure 3.

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