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mathematics
college algebra graphs and models

Questions and Answers of College Algebra Graphs And Models

Describe the determinants Dx and Dy in terms of the coefficients and constants in a system of two equations in two variables.
If you are given two matrices, A and B, explain how to determine if B is the multiplicative inverse of A.
Explain how to evaluate a third-order determinant.
Explain why a matrix that does not have the same number of rows and columns cannot have a multiplicative inverse.
If your graphing utility has a ref (row-echelon form) command or a rref (reduced row-echelon form) command, use this feature to verify your work with any five systems that you solved from Exercises
In a certain county, the proportion of voters in each age group registered as Republicans, Democrats, or Independents is given by the following matrix, which we’ll call A.The distribution, by age
Completing the transition to adulthood is measured by one or more of the following: leaving home, finishing school, getting married, having a child, or being financially independent. The bar graph
Each column in the matrix describes a point on the letter. The order of the columns shows the direction in which a pencil must move to draw the letter. The L is completed by connecting the last point
Each column in the matrix describes a point on the letter. The order of the columns shows the direction in which a pencil must move to draw the letter. The L is completed by connecting the last point
Use the feature of your graphing utility that evaluates the determinant of a square matrix to verify any five of the determinants that you evaluated by hand in Exercises 1–10, 23–28, or
The table gives an estimate of basic caloric needs for different age groups and activity levels.a. Use a 3 × 3 matrix to represent the daily caloric needs, by age and activity level, for men. Call
Solve using a graphing utility’s ref or rref command: 2x12x₂ + 3x3 - X4 x₁ + 2x₂x3 + 2x4 - x₁ + -x₁ + x₂ x2 x1 - x₂ x2 - - = 12 x5 X5 = -7 x3 + x4 - 5x5 1 3x5 0 X4 + X5 = 4. x3 - 2x4
In Exercises 65–70, use a graphing utility to find the multiplicative inverse of each matrix. Check that the displayed inverse is correct. -4 6 1 -2
When expanding a determinant by minors, when is it necessary to supply minus signs?
Explain how to find the multiplicative inverse for a 2 × 2 invertible matrix.
In Exercises 64–65, use a graphing utility to evaluate the determinant for the given matrix. 3 -2 1 352 -5 -1 43 -1 4 27 5 0 -6 5
The final grade in a particular course is determined by grades on the midterm and final. The grades for five students and the two grading systems are modeled by the following matrices. Call the first
In applying Cramer’s Rule, what should you do if D = 0?
Explain how to write a linear system of three equations in three variables as a matrix equation.
In Exercises 65–70, use a graphing utility to find the multiplicative inverse of each matrix. Check that the displayed inverse is correct. 3 -1 -2 1
In Exercises 63–66, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.The augmented matrix for the system x - 3y
In Exercises 59–62, determine whether each statement makes sense or does not make sense, and explain your reasoning.When I use matrices to solve linear systems, the only arithmetic involves
The process of solving a linear system in three variables using Cramer’s Rule can involve tedious computation. Is there a way of speeding up this process, perhaps using Cramer’s Rule to find the
In Exercises 64–65, use a graphing utility to evaluate the determinant for the given matrix. 82 20 -3 633 6 -1 0 4 7 6 -5 5 3 2 1 -1 2 -3 1 45 -2 -1 -8
Explain how to solve the matrix equation AX = B.
The table shows the daily production level and profit for a business.Use the quadratic function y = ax2 + bx + c to determine the number of units that should be produced each day for maximum profit.
In Exercises 59–62, determine whether each statement makes sense or does not make sense, and explain your reasoning.When I use matrices to solve linear systems, I spend most of my time using row
If you could use only one method to solve linear systems in three variables, which method would you select? Explain why this is so.
What is a cryptogram?
In Exercises 59–62, determine whether each statement makes sense or does not make sense, and explain your reasoning.Using row operations on an augmented matrix, I obtain a row in which 0s appear to
a.b.c.d. Describe the pattern in the given determinants.e. Describe the pattern in the evaluations. Evaluate: a 0 a a
It’s January 1, and you’ve written down your major goal for the year. You do not want those closest to you to see what you’ve written in case you do not accomplish your objective. Consequently,
In Exercises 65–70, use a graphing utility to find the multiplicative inverse of each matrix. Check that the displayed inverse is correct. -2 -5 3 1 2 -1 1 -1
In Exercises 63–66, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.A matrix row operation such as - 4/5 R1 +
In Exercises 65–70, use a graphing utility to find the multiplicative inverse of each matrix. Check that the displayed inverse is correct. 1 -3 3 1 12 2 -3 -1 -1 2
In Exercises 65–70, use a graphing utility to find the multiplicative inverse of each matrix. Check that the displayed inverse is correct. 7 -2 4 -1 -3 0 1 0 0 1 10-1 2 -1 -2
In Exercises 63–66, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.In solving a linear system of three
What is meant by the order of a matrix? Give an example with your explanation.
Graph the solution set of the system: x + y ≤ 7 x + 4y > -8.
What is the fastest method for solving a linear system with your graphing utility?
Exercises 71–73 will help you prepare for the material covered in the next section. In each exercise, refer to the following system:Show that (12z + 1, 10z - 1, z) satisfies the system for z = 0.
In Exercises 65–70, use a graphing utility to find the multiplicative inverse of each matrix. Check that the displayed inverse is correct. 1 0 1 4 20 0 1 30 002 0 0 1
In Exercises 63–66, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.The row operation kRi + Rj indicates that
What does aij mean?
In Exercises 67–70, determine whether each statement makes sense or does not make sense, and explain your reasoning.I’m solving a linear system using a determinant that contains two rows and
What are equal matrices?
In Exercises 71–76, write each system in the form AX = B. Then solve the system by entering A and B into your graphing utility and computing A-1 B. y +z = -6 X 4x + 2y + z = 4x - 2y + z 9 2y + z =
In Exercises 67–70, determine whether each statement makes sense or does not make sense, and explain your reasoning.I can speed up the tedious computations required by Cramer’s Rule by using the
Exercises 71–73 will help you prepare for the material covered in the next section. In each exercise, refer to the following system:Show that (12z + 1, 10z - 1, z) satisfies the system for z = 1.
How are matrices added?
In Exercises 71–76, write each system in the form AX = B. Then solve the system by entering A and B into your graphing utility and computing A-1 B. 0 1 -x + y 2xy + z = −1 y + 2z= ||
Describe how to subtract matrices.
Consider the systemUse Cramer’s Rule to prove that if the first equation of the system is replaced by the sum of the two equations, the resulting system has the same solution as the original
In Exercises 67–70, determine whether each statement makes sense or does not make sense, and explain your reasoning.Using Cramer’s Rule to solve a linear system, I found the value of D to be
In Exercises 71–76, write each system in the form AX = B. Then solve the system by entering A and B into your graphing utility and computing A-1 B. 3x-2y + z = -2 4x - 5y + 3z = :-9 2x у + 5z = -5
Describe matrices that cannot be added or subtracted.
In Exercises 71–76, write each system in the form AX = B. Then solve the system by entering A and B into your graphing utility and computing A-1 B. V W -3x + X v + w x + v+w+ y 4y + z = -3 = -1 + z
Describe how to perform scalar multiplication. Provide an example with your description.
In Exercises 71–76, write each system in the form AX = B. Then solve the system by entering A and B into your graphing utility and computing A-1 B. w + x + y + z = 4 w + 3x - 2y + 2z = 7 2w + 2x +
Describe how to multiply matrices.
What happens to the value of a second-order determinant if the two columns are interchanged?
Describe when the multiplication of two matrices is not defined.
Low-resolution digital photographs use 262,144 pixels in a 512 × 512 grid. If you enlarge a low-resolution digital photograph enough, describe what will happen.
We have seen that determinants can be used to solve linear equations, give areas of triangles in rectangular coordinates, and determine equations of lines. Not impressed with these applications?
In Exercises 77–80, determine whether each statement makes sense or does not make sense, and explain your reasoning.I added matrices of the same order by adding corresponding elements.
In Exercises 13–26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. 16 x² 36 1
In Exercises 12–18, graph each equation.x2 + 4(y - 1)2 = 4
In Exercises 1–18, graph each ellipse and locate the foci.7x2 = 35 - 5y2
In Exercises 17–30, find the standard form of the equation of each parabola satisfying the given conditions.Focus: (7, 0); Directrix: x = -7
In Exercises 15–22, graph each hyperbola. Locate the foci and find the equations of the asymptotes.9x2 - 16y2 = 144
17. Express the sum using summation notation. Use i for the index of summation. 1 2 + 3 4 3 + + 5 + 18 20
Expand using logarithmic properties. Where possible, evaluate logarithmic expressions. logs x³√y 125
In Exercises 15–22, graph each hyperbola. Locate the foci and find the equations of the asymptotes. y² 16 - x² = 1
In Exercises 5–16, find the focus and directrix of the parabola with the given equation. Then graph the parabola.8y2 + 4x = 0
In Exercises 1–18, graph each ellipse and locate the foci.4x2 + 25y2 = 100
In Exercises 12–18, graph each equation.(x + 1)2 + (y - 1)2 = 4
In Exercises 13–26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. 2 X 144 81 = 1
In Exercises 13–26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. x² 100 y² 64 1
In Exercises 12–18, graph each equation.x2 + 4y2 = 4
Express 0.4̅5̅ as a fraction in lowest terms.
If f(x) = x2 - 4 and g(x) = x + 2, find (g ° f)(x).
In Exercises 1–18, graph each ellipse and locate the foci.4x2 + 16y2 = 64
a. List all possible rational roots of 32x3 - 52x2 + 17x + 3 = 0.b. The graph of f(x) =32x3 - 52x2 + 17x + 3 is shown in a [-1, 3, 1] by [-2, 6, 1] viewing rectangle. Use the graph of f and synthetic
In Exercises 5–16, find the focus and directrix of the parabola with the given equation. Then graph the parabola.x2 - 6y = 0
In Exercises 13–26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. 2 X 16 25 1
The figure shows the graph of y = f(x) and its two vertical asymptotes.a. Find the domain and the range of f.b. What is the relative minimum and where does it occur?c. Find the interval on which f is
In Exercises 12–18, graph each equation.x2 - y2 = 4
In Exercises 12–15, find each indicated sum. 00 1=1 2 5 1-1
In Exercises 1–18, graph each ellipse and locate the foci.9x2 + 4y2 = 36
In Exercises 15–22, graph each hyperbola. Locate the foci and find the equations of the asymptotes. x2 – y2 = 1 ا 16
An elliptical pool table has a ball placed at each focus. If one ball is hit toward the edge of the table, explain what will occur.
In Exercises 12–15, find each indicated sum. 6 i=1 3) i
In Exercises 12–15, find each indicated sum. 50 Σ (3i – 2) i=1
In Exercises 11–13, graph each equation, function, or system in a rectangular coordinate system. 5x + y ≤ 10 1 y = -x + 2 4
In Exercises 1–18, graph each ellipse and locate the foci.y2 = 1 - 4x2
In Exercises 12–18, graph each equation.x2 + y2 = 4
In Exercises 5–12, find the standard form of the equation of each hyperbola satisfying the given conditions.Center: (-2, 1); Focus: (-2, 6); vertex: (-2, 4)
A semielliptic archway has a height of 15 feet at the center and a width of 50 feet, as shown in the figure. The 50-foot width consists of a two-lane road. Can a truck that is 12 feet high and 14
In Exercises 13–26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. X 9 1² 25 1
In Exercises 5–16, find the focus and directrix of the parabola with the given equation. Then graph the parabola.y2 - 6x = 0

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