?(Calculus required) Use the method of Example 7 to approximate the

In each part, determine whether the equation is linear in \(x_{1}, x_{2}\), and \(x_{3}\) a. \(x_{1}+5 x_{2}-\sqrt{2 x_{3}}=1\) b. \(x_{1}+3 x_{2}+x_{1} x_{3}=2\) c. \(x_{1}=-7 x_{2}+3 x_{3}\) d. \(x_{1}^{-2}+x_{2}+8 x_{3}=5\) e. \(x_{1}^{3 / 5}-2 x_{2}+x_{3}=4\) f. \(\pi x_{1}-\sqrt{2 x_{2}}=7^{1 / 3}\) Equation Transcription: Text Transcription: x_{1}, x_{2} x_3 x_1 + 5x_2 - sqrt{2x_3} = 1 x_1 + 3x_2 + x_{1} x_{3} = 2 x_1 = -7x_2 + 3x_3 x_1^-2 + x_2 + 8x_3 = 5 x_1^? - 2x_2 + x_3 = 4 pi x_1 - sqrt{2x_2} = 7^1/3

In each part, determine whether the equation is linear in \(x\) and \(y\). a. \(2^{1 / 3} x+\sqrt{3 y}=1\) b. \(2 x^{1 / 3}+\sqrt{3 y}=1\) c. \(\cos \left(\frac{\pi}{7}\right) x-4 y=\log 3\) d. \(\frac{\pi}{7} \cos x-4 y=0\) e. \(x y=1\) f. \(y+7=x\) Equation Transcription: Text Transcription: x y 2^{1/3}x + sqrt{3y} = 1 2x^? + sqrt{3y} = 1 cos (frac{pi}{7})x - 4y = log 3 frac{pi}{7} cos x - 4y = 0 xy = 1 y + 7 = x

Using the notation of Formula (7), write down a general linear system of a. two equations in two unknowns. b. three equations in three unknowns. c. two equations in four unknowns.

Write down the augmented matrix for each of the linear systems in Exercise 3 .

In each part of Exercises , find a system of linear equations in the unknowns \(x_{1}, x_{2}, x_{3}\),......, that corresponds to the given augmented matrix. a.\(\left[\begin{array}{cccc} & 2 & 0 & 0 & \\& 3 & -4 & 0 \\& 0 & 1 & 1\end{array}\right]\) b. \(\left[\begin{array}{cccc}3 & 0 & -2 & 5 \\7 & 1 & 4 & -3 \\0 & -2 & 1 & 7\end{array}\right]\) Equation Transcription: [] [] Text Transcription: x_{1}, x_{2}, x_{3} [ 2 & 0 & 0 & & 3 & -4 & 0 & 0 & 1 & 1] [3 & 0 & -2 & 5 7 & 1 & 4 & -3 0 & -2 & 1 & 7]

In each part of Exercises , find a system of linear equations in the unknowns \(x_{1}, x_{2}, x_{3}\),......, that corresponds to the given augmented matrix. a.\(\left[\begin{array}{ccccc}0 & 3 & -1 & -1 & -1 \\5 & 2 & 0 & -3 & -6\end{array}\right]\) b.\(\left[\begin{array}{cccccc}3&0&1&-4&3\\-4&0&4&1&-3\\1&3&0&-2&-9\\0&0&0&-1&-2\end{array}\right]\) Equation Transcription: Text Transcription: x_{1}, x_{2}, x_{3} [0 & 3 & -1 & -1 & -1 \\5 & 2 & 0 & -3 & -6] [3&0&1&-4&3\\-4&0&4&1&-3\\1&3&0&-2&-9\\0&0&0&-1&-2]

In each part of Exercises , find the augmented matrix for the linear system. a. \(-2 x_{1}=6\) \(3 x_{1}=8\) \(9 x_{1}=-3\) b. \(6 x_{1}-x_{2}+3 x_{3}=4\) \(5 x_{2}-x_{3}=1\) c. \(2 x_{2}-3 x_{4}+x_{5}=0\)\ \(-3 x_{1}-x_{2}+x_{3}=-1\) \(6 x_{1}+2 x_{2}-x_{3}+2 x_{4}=3 x_{5}=6\) Equation Transcription: Text Transcription: -2x_1 = 6 3x_1 = 8 9x_1 = -3 6x_1 - x_2 + 3x_3 = 4 5x_2 - x_3 = 1 2x_2 - 3x_4 + x_5 = 0 -3x_1 - x_2 + x_3 = -1 6x_1 + 2x_2 - x_3 + 2x_4 = 3x_5 = 6

In each part of Exercises , find the augmented matrix for the linear system. a. \(3 x_{1}-2 x_{2}=-1\) \(3 x_{1}-x_{2}+4 x_{3}=7\) \(7 x_{1}+3 x_{2}=2\) b. \(2 x_{1}+2 x_{3}=1\) \(3 x_{1}-x_{2}+4 x_{3}=7\) \(6 x_{1}+x_{2}-x_{3}=0\) c. \(x_{1}=1\) \(x_{2}=2\) \(x_{3}=3\) Equation Transcription: Text Transcription: 3x_1 - 2x_2 = -1 3x_1 - x_2 + 4x_3 = 7 7x_1 + 3x_2 = 2 2 x_1 + 2x_3 = 1 3x_1 - x_2 + 4x_3 = 7 6 x_1 + x_2 - x_3 = 0 x_1 = 1 x_2 = 2 x_3 = 3

In each part, determine whether the given 3 -tuple is a solution of the linear system \(2 x_{1}-4 x_{2}-x_{3}=1\) \(x_{1}-3 x_{2}+x_{3}=1\) \(3 x_{1}-5 x_{2}-3 x_{3}=1\) a. (3, 1, 1) b. (3, -1, 1) c. (13, 5, 2) d. (\(\frac{13}{2}, \frac{5}{2}, 2\)) e. (17, 7, 5) Equation Transcription: Text Transcription: 2x_1 - 4x_2 - x_3 = 1 x_1 - 3x_2 + x_3 = 1 3x_1 - 5x_2 - 3x_3 = 1 frac{13}{2}, frac{5}{2}, 2

In each part, determine whether the given 3 -tuple is a solution of the linear system \( x+2 y-2 z=3\) \(3 x-y+z=1\) \(-x+5 y-5 z=5\) a. \(\left(\frac{5}{7}, \frac{8}{7}, 1\right)\) b. \(\left(\frac{5}{7}, \frac{8}{7}, 0\right)\) c. (5,8,1) d. \(\left(\frac{5}{7}, \frac{10}{7}, \frac{2}{7}\right)\) e. \(\left(\frac{5}{7}, \frac{22}{7}, 2\right)\) Equation Transcription: Text Transcription: x+2 y-2 z=3\) 3 x-y+z=1\) -x+5 y-5 z=5\) frac{5}{7}, frac{8}{7}, 1) frac{5}{7}, frac{8}{7}, 0) frac{5}{7}, frac{10}{7}, frac{2}{7}) frac{5}{7}, frac{22}{7}, 2)

In each part, solve the linear system, if possible, and use the result to determine whether the lines represented by the equations in the system have zero, one, or infinitely many points of intersection. If there is a single point of intersection, give its coordinates, and if there are infinitely many, find parametric equations for them. a. \(3 x-2 y=4\) b. \(2 x-4 y=1\) c.\(x-2 y=0\) \(6 x-4 y=94 x-8 y=2 x-4 y=8\) Equation Transcription: Text Transcription: 3 x-2 y=4 2 x-4 y=1 x-2 y=0 6 x-4 y=94 x-8 y=2 x-4 y=8

Under what conditions on \(a\) and \(b\) will the linear system have no solutions, one solution, infinitely many solutions? \(2 x-3 y=a 4 x-6 y=b\) Equation Transcription: Text Transcription: a b 2x - 3y = a 4x - 6y = b

In each part of Exercises 13-14, use parametric equations to describe the solution set of the linear equation. a. \(7 x-5 y=3$\) b. \(3 x_{1}-5 x_{2}+4 x_{3}=7\) c. \(-8 x_{1}+2 x_{2}-5 x_{3}+6 x_{4}=1\) d. \(3 v-8 w+2 x-y+4 z=0\) Equation Transcription: Text Transcription: 7x -5y = 3 3x_1 - 5x_2 + 4x_3 = 7 -8x_1 + 2x_2 - 5x_3 + 6x_4 = 1 3v - 8w + 2x - y + 4z = 0

In each part of Exercises 13-14, use parametric equations to describe the solution set of the linear equation. a. \(x+10 y=2\) b. \(x_{1}+3 x_{2}-12 x_{3}=3\) c. \(4 x_{1}+2 x_{2}+3 x_{3}+x_{4}=20\) d. \(v+w+x-5 y+7 z=0\) Equation Transcription: Text Transcription: x + 10y = 2 x_1 + 3x_2 - 12x_3 = 3 4x_1 + 2x_2 + 3x_3 + x_4 = 20 v + w + x - 5y + 7z= 0

In Exercises 15-16, each linear system has infinitely many solutions. Use parametric equations to describe its solution set. a. \(2 x-3 y=1\) \(6 x-9 y=3\) b. \(x_{1}+3 x_{2}-x_{3}= -4\) \(3 x_{1}+9 x_{2}-3 x_{3}=-12\) \(-x_{1}-3 x_{2}+x_{3}=4\) Equation Transcription: Text Transcription: 2x - 3y = 1 6x - 9y = 3 x_1 + 3x_2 - x_3 = -4 3x_1 + 9x_2 - 3x_3 = -12 -x_1 - 3x_2 + x_3 = 4

In Exercises 15-16, each linear system has infinitely many solutions. Use parametric equations to describe its solution set. a. \(6 x_{1}+2 x_{2}=-8\) \(3 x_{1}+x_{2}=-4\) b. \(2 x-y+2 z=-4\) \(6 x-3 y+6 z= -12\) \(-4 x+2 y-4 z= 8\) Equation Transcription: Text Transcription: 6x_1 + 2x_2 = -8 3x_1 + x_2 = -4 2x - y + 2z = -4 6x - 3y + 6z = -12 -4x + 2y - 4z = 8

In Exercises 17-18, find a single elementary row operation that will create a 1 in the upper left corner of the given augmented matrix and will not create any fractions in its first row. a. \(\left[\begin{array}{rrrr}-3 & -1 & 2 & 4 \\ 2 & -3 & 3 & 2 \\ 0 & 2 & -3 & 1\end{array}\right]\) b. \(\left[\begin{array}{rrrr}0 & -1 & -5 & 0 \\ 2 & -9 & 3 & 2 \\ 1 & 4 & -3 & 3\end{array}\right]\) Equation Transcription: [] [] Text Transcription: [-3 & -1 & 2 & 4 \\ 2 & -3 & 3 & 2 \\ 0 & 2 & -3 & 1] [0 & -1 & -5 & 0 \\ 2 & -9 & 3 & 2 \\ 1 & 4 & -3 & 3]

In Exercises 17-18, find a single elementary row operation that will create a 1 in the upper left corner of the given augmented matrix and will not create any fractions in its first row. a. \(\left[\begin{array}{rrrr}2 & 4 & -6 & 8 \\ 7 & 1 & 4 & 3 \\ -5 & 4 & 2 & 7\end{array}\right]\) b. \(\left[\begin{array}{rrrr}7 & -4 & -2 & 2 \\ 3 & -1 & 8 & 1 \\ -6 & 3 & -1 & 4\end{array}\right]\) Equation Transcription: [] [] Text Transcription: [2 & 4 & -6 & 8 \\ 7 & 1 & 4 & 3 \\ -5 & 4 & 2 & 7] [7 & -4 & -2 & 2 \\ 3 & -1 & 8 & 1 \\ -6 & 3 & -1 & 4]

In Exercises 19-20, find all values of \(k\) for which the given aug. mented matrix corresponds to a consistent linear system. a. \(\left[\begin{array}{rrr}1 & k & -4 \\ 4 & 8 & 2\end{array}\right]\) b. \(\left[\begin{array}{lll}1 & k & -1 \\ 4 & 8 & -4\end{array}\right]\) Equation Transcription: Text Transcription: [1 & k & -4 \\ 4 & 8 & 2] [1 & k & -1 \\ 4 & 8 & -4]

In Exercises 19-20, find all values of \(k\) for which the given aug. mented matrix corresponds to a consistent linear system. a. \(\left[\begin{array}{rrr}3 & -4 & k \\ -6 & 8 & 5\end{array}\right]\) b. \(\left[\begin{array}{rrr}k & 1 & -2 \\ 4 & -1 & 2\end{array}\right]\) Equation Transcription: Text Transcription: [3 & -4 & k \\ -6 & 8 & 5] [k & 1 & -2 \\ 4 & -1 & 2]

The curve \(y=a x^{2}+b x+c\) shown in the accompanying figure passes through the points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\), and \(\left(x_{3}, y_{3}\right)\). Show that the coefficients \(a, b\), and \(c\) form a solution of the system of linear equations whose augmented matrix is \(\left[\begin{array}{llll}x_{1}^{2} & x_{1} & 1 & y_{1} \\x_{2}^{2} & x_{2} & 1 & y_{2} \\x_{3}^{2} & x_{3} & 1 & y_{3}\end{array}\right]\) Equation Transcription: [] Text Transcription: y = a x^2 +bx + c (x_1, y_1),(x_2, y_2}) (x_3, y_3) a, b, c [x_{1}^{2} & x_{1} & 1 & y_{1} \\x_{2}^{2} & x_{2} & 1 & y_{2} \\x_{3}^{2} & x_{3} & 1 & y_{3}]

Explain why each of the three elementary row operations does not affect the solution set of a linear system.

Show that if the linear equations \(x_{1}+k x_{2}=c\) and \(x_{1}+l x_{2}=d\) have the same solution set, then the two equations are identical (i.e., \(k = l\) and \(c = d\) ). Equation Transcription: Text Transcription: x_1 + kx_2 = c x_1 + lx_2 = d k = l c = d

Consider the system of equations \(ax\ +\ by=k\ cx+dy\ =\ l\ ex\ +\ fy\ =\ m\) Discuss the relative positions of the lines \(ax\ +\ by=k\ cx+dy\ =\ l\), and \(ex\ +\ fy\ =\ m\) when a. the system has no solutions. b. the system has exactly one solution. c. the system has infinitely many solutions. Equation Transcription: Text Transcription: ax + by = k cx + dy = l ex + fy = m ax + by = k cx + dy = l ex + fy = m

Suppose that a certain diet calls for 7 units of fat, 9 units of protein, and 16 units of carbohydrates for the main meal, and suppose that an individual has three possible foods to choose from to meet these requirements: Food 1: Each ounce contains 2 units of fat, 2 units of protein, and 4 units of carbohydrates. Food 2: Each ounce contains 3 units of fat, 1 unit of protein, and 2 units of carbohydrates. Food 3: Each ounce contains 1 unit of fat, 3 units of protein, and 5 units of carbohydrates. Let \(x, y\), and \(z\) denote the number of ounces of the first, second, and third foods that the dieter will consume at the main meal. Find (but do not solve) a linear system in \(x, y\), and \(z\) whose solution tells how many ounces of each food must be consumed to meet the diet requirements. Equation Transcription: Text Transcription: x,y z x,y z

Suppose that you want to find values for \(a, b\), and \(c\) such that the parabola \(y=a x^{2}+b x+c\) passes through the points (1, 1), (2, 4), and (-1, 1). Find (but do not solve) a system of linear equations whose solutions provide values for \(a, b\), and \(c\). How many solutions would you expect this system of equations to have, and why? Equation Transcription: Text Transcription: a, b c y = ax^2 + bx + c a, b c

Suppose you are asked to find three real numbers such that the sum of the numbers is 12 , the sum of two times the first plus the second plus two times the third is 5, and the third number is one more than the first. Find (but do not solve) a linear system whose equations describe the three conditions.

In parts (a)–(h) determine whether the statement is true or false, and justify your answer. A linear system whose equations are all homogeneous must be consistent.

In parts (a)–(h) determine whether the statement is true or false, and justify your answer. Multiplying a row of an augmented matrix through by zero is an acceptable elementary row operation.

In parts (a)–(h) determine whether the statement is true or false, and justify your answer. The linear system \(x ? y = 3\) \(2x ? 2y = k\) cannot have a unique solution, regardless of the value of \(k\). Equation Transcription: Text Transcription: x ? y = 3 2x ? 2y = k k

In parts (a)–(h) determine whether the statement is true or false, and justify your answer. A single linear equation with two or more unknowns must have infinitely many solutions.

In parts (a)–(h) determine whether the statement is true or false, and justify your answer. If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent.

In parts (a)–(h) determine whether the statement is true or false, and justify your answer. If each equation in a-consistent linear system is multiplied through by a constant \(c\), then all solutions to the new system can be obtained by multiplying solutions from the original system by \(c\). Equation Transcription: c Text Transcription: c c

In parts (a)–(h) determine whether the statement is true or false, and justify your answer. Elementary row operations permit one row of an augmented matrix to be subtracted from another.

In parts (a)–(h) determine whether the statement is true or false, and justify your answer. The linear system with corresponding augmented matrix \(\left[\begin{array}{ccc}2 & -1 & 4 \\0 & 0 & -1\end{array}\right]\) is consistent. Equation Transcription: Text Transcription: [2 & -1 & 4 \\0 & 0 & -1]

Solve the linear systems in Examples 2, 3, and 4 to see how your technology utility handles the three types of systems.

Use the result in Exercise 21 to find values of \(a, b\), and \(c\) for which the curve \(y=a x^{2}+b x+c\) passes through the points (?1, 1, 4), (0, 0, 8), and (1, 1, 7). Equation Transcription: Text Transcription: a, b c y = ax^2 + bx + c

The accompanying figure shows a network in which the flow rate and direction of flow in certain branches are known. Find the flow rates and directions of flow in the remaining branches.

The accompanying figure shows known flow rates of hydrocarbons into and out of a network of pipes at an oil refinery. a. Set up a linear system whose solution provides the unknown flow rates. b. Solve the system for the unknown flow rates. c. Find the flow rates and directions of flow if \(\mathrm{x}_{4}=50\) and \(x_{6}=0\). Equation Transcription: Text Transcription: x_4=50 x_6=0

The accompanying figure shows a network of one-way streets with traffic flowing in the directions indicated. The flow rates along the streets are measured as the average number of vehicles per hour. a. Set up a linear system whose solution provides the unknown flow rates. b. Solve the system for the unknown flow rates. c. If the flow along the road from \(A \text { to } B\) must be reduced for construction, what is the minimum flow that is required to keep traffic flowing on all roads? Equation Transcription: Text Transcription: A to B x1 x2 x3 x4

The accompanying figure shows a network of one-way streets with traffic flowing in the directions indicated. The flow rates along the streets are measured as the average number of vehicles per hour. a. Set up a linear system whose solution provides the unknown flow rates. b. Solve the system for the unknown flow rates. c. Is it possible to close the road from \(A \text { to } B\) for construction and keep traffic flowing on the other streets? Explain. Equation Transcription: Text Transcription: A to B x1 x2 x3 X4 x5

In Exercises 5–8, analyze the given electrical circuits by finding the unknown currents. Equation Transcription: Text Transcription: 8V 2 \omega 6V 4 \omega I_1 2 \omega I_2 I_3

In Exercises 5–8, analyze the given electrical circuits by finding the unknown currents. Equation Transcription: Text Transcription: +2V 6 \omega 1V+ 4 \omega 2 \omega I_1 I_2 I_3

In Exercises 5–8, analyze the given electrical circuits by finding the unknown currents. Equation Transcription: Text Transcription: 10 V 20 \omega 20 \omega 20 \omega 20 \omega 10 V I_1 I_2 I_3 I_4 I_5 I_6

In Exercises 5–8, analyze the given electrical circuits by finding the unknown currents. Equation Transcription: Text Transcription: 5 V 4 V 3 V 3 \omega I_1 4 \omega I_2 5 \omega I_3

In Exercises 9-12, write a balanced equation for the given chemical reaction. \(\mathrm{C}_{3} \mathrm{H}_{8}+\mathrm{O}_{2} \rightarrow \mathrm{CO}_{2}+\mathrm{H}_{2} \mathrm{O}\) [propane combustion] Equation Transcription: Text Transcription: C_3 H_8 + O_2 \rightarrow CO_2 + H2_O

In Exercises 9-12, write a balanced equation for the given chemical reaction. \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6} \rightarrow \mathrm{CO}_{2}+\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\) Equation Transcription: Text Transcription: C_6 H_12 O_6 \rightarrow CO_2 + C_2 H_5 OH

In Exercises 9-12, write a balanced equation for the given chemical reaction. \(\mathrm{CH}_{3} \mathrm{COF}+\mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{CH}_{3} \mathrm{COOH}+\mathrm{HF}\) Equation Transcription: Text Transcription: CH_3 COF + H_2O \rightarrow CH_3COOH+HF

In Exercises 9-12, write a balanced equation for the given chemical reaction. \(\mathrm{CO}_{2}+\mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}+\mathrm{O}_{2}\) [photosynthesis] Equation Transcription: Text Transcription: CO_2 + H_2O \rightarrow C_6 H_12 O_6 + O_2

Find the quadratic polynomial whose graph passes through the points \((1,1),(2,2), \text { and }(3,5)\). Equation Transcription: Text Transcription: (1,1),(2,2), and (3,5)

Find the quadratic polynomial whose graph passes through the points \((0,0),(-1,1), \text { and }(1,1)\) Equation Transcription: Text Transcription: (0,0),(-1,1), and (1,1)

Find the cubic polynomial whose graph passes through the points \((-1,-1),(0,1),(1,3),(4,-1)\) Equation Transcription: Text Transcription: (-1,-1),(0,1),(1,3),(4,-1)

The accompanying figure shows the graph of a cubic polynomial. Find the polynomial.

a. Find an equation that represents the family of all second degree polynomials that pass through the points \((0,1) \text { and }(1,2)\). [Hint: The equation will involve one arbitrary parameter that produces the members of the family when varied.] b. By hand, or with the help of a graphing utility, sketch four curves in the family. Equation Transcription: Text Transcription: (0,1) and (1,2)

In this section we have selected only a few applications of linear systems. Using the Internet as a search tool, try to find some more real-world applications of such systems. Select one that is of interest to you and write a paragraph about it.

In parts (a)–(e) determine whether the statement is true or false, and justify your answer. a. In any network, the sum of the flows out of a node must equal the sum of the flows into a node.

In parts (a)–(e) determine whether the statement is true or false, and justify your answer. b. When a current passes through a resistor, there is an increase in the electrical potential in a circuit.

In parts (a)–(e) determine whether the statement is true or false, and justify your answer. c. Kirchhoff’s current law states that the sum of the currents flowing into a node equals the sum of the currents flowing out of the node.

In parts (a)–(e) determine whether the statement is true or false, and justify your answer. d. A chemical equation is called balanced if the total number of atoms on each side of the equation is the same.

In parts (a)–(e) determine whether the statement is true or false, and justify your answer. e. Given any n points in the \(x y \text { - plane }\), there is a unique polynomial of degree \(n-1\) or less whose graph passes through those points. Equation Transcription: Text Transcription: xy-plane n-1

The following table shows the lifting force on an aircraft wing measured in a wind tunnel at various wind velocities. Model the data with an interpolating polynomial of degree 5, and use that polynomial to estimate the lifting force at \(2000 \mathrm{ft} / \mathrm{s}\). Equation Transcription: Text Transcription: 2000 ft/s

(Calculus required) Use the method of Example 7 to approximate the integral \(\int_{0}^{1} e^{x^{2}} d x\) by subdividing the interval of integration into five equal parts and using an interpolating polynomial to approximate the integrand. Compare your answer to that obtained using the numerical integration capability of your technology utility. Equation Transcription: Text Transcription: \int_0^1 e^x^2 d x

Use the method of Example 5 to balance the chemical equation. \(\mathrm{Fe}_{2} \mathrm{O}_{3}+\mathrm{Al} \rightarrow \mathrm{Al}_{2} \mathrm{O}_{3}+\mathrm{Fe}\) (Fe = iron, Al = aluminum, O = oxygen) Equation Transcription: Text Transcription: Fe_2 O_3 + Al \rightarrow Al_2 O_3 + F_e

Determine the currents in the accompanying circuit. Equation Transcription: Text Transcription: I_1 I_2 I_3 I_1 I_2 I_3 20 V 12 V 3 \omega 470 \omega 2 \omega

The accompanying figure shows a network in which the flow rate and direction of flow in certain branches are known. Find the flow rates and directions of flow in the remaining branches.

The accompanying figure shows known flow rates of hydrocarbons into and out of a network of pipes at an oil refinery. a. Set up a linear system whose solution provides the unknown flow rates. b. Solve the system for the unknown flow rates. c. Find the flow rates and directions of flow if \(\mathrm{x}_{4}=50\) and \(x_{6}=0\). Equation Transcription: Text Transcription: x_4=50 x_6=0

The accompanying figure shows a network of one-way streets with traffic flowing in the directions indicated. The flow rates along the streets are measured as the average number of vehicles per hour. a. Set up a linear system whose solution provides the unknown flow rates. b. Solve the system for the unknown flow rates. c. If the flow along the road from \(A \text { to } B\) must be reduced for construction, what is the minimum flow that is required to keep traffic flowing on all roads? Equation Transcription: Text Transcription: A to B x1 x2 x3 x4

The accompanying figure shows a network of one-way streets with traffic flowing in the directions indicated. The flow rates along the streets are measured as the average number of vehicles per hour. a. Set up a linear system whose solution provides the unknown flow rates. b. Solve the system for the unknown flow rates. c. Is it possible to close the road from \(A \text { to } B\) for construction and keep traffic flowing on the other streets? Explain. Equation Transcription: Text Transcription: A to B x1 x2 x3 X4 x5

In Exercises 5–8, analyze the given electrical circuits by finding the unknown currents. Equation Transcription: Text Transcription: 8V 2 \omega 6V 4 \omega I_1 2 \omega I_2 I_3

In Exercises 5–8, analyze the given electrical circuits by finding the unknown currents. Equation Transcription: Text Transcription: +2V 6 \omega 1V+ 4 \omega 2 \omega I_1 I_2 I_3

In Exercises 5–8, analyze the given electrical circuits by finding the unknown currents. Equation Transcription: Text Transcription: 10 V 20 \omega 20 \omega 20 \omega 20 \omega 10 V I_1 I_2 I_3 I_4 I_5 I_6

In Exercises 5–8, analyze the given electrical circuits by finding the unknown currents. Equation Transcription: Text Transcription: 5 V 4 V 3 V 3 \omega I_1 4 \omega I_2 5 \omega I_3

In Exercises 9-12, write a balanced equation for the given chemical reaction. \(\mathrm{C}_{3} \mathrm{H}_{8}+\mathrm{O}_{2} \rightarrow \mathrm{CO}_{2}+\mathrm{H}_{2} \mathrm{O}\) [propane combustion] Equation Transcription: Text Transcription: C_3 H_8 + O_2 \rightarrow CO_2 + H2_O

In Exercises 9-12, write a balanced equation for the given chemical reaction. \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6} \rightarrow \mathrm{CO}_{2}+\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\) Equation Transcription: Text Transcription: C_6 H_12 O_6 \rightarrow CO_2 + C_2 H_5 OH

In Exercises 9-12, write a balanced equation for the given chemical reaction. \(\mathrm{CH}_{3} \mathrm{COF}+\mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{CH}_{3} \mathrm{COOH}+\mathrm{HF}\) Equation Transcription: Text Transcription: CH_3 COF + H_2O \rightarrow CH_3COOH+HF

In Exercises 9-12, write a balanced equation for the given chemical reaction. \(\mathrm{CO}_{2}+\mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}+\mathrm{O}_{2}\) [photosynthesis] Equation Transcription: Text Transcription: CO_2 + H_2O \rightarrow C_6 H_12 O_6 + O_2

Find the quadratic polynomial whose graph passes through the points \((1,1),(2,2), \text { and }(3,5)\). Equation Transcription: Text Transcription: (1,1),(2,2), and (3,5)

Find the quadratic polynomial whose graph passes through the points \((0,0),(-1,1), \text { and }(1,1)\) Equation Transcription: Text Transcription: (0,0),(-1,1), and (1,1)

Find the cubic polynomial whose graph passes through the points \((-1,-1),(0,1),(1,3),(4,-1)\) Equation Transcription: Text Transcription: (-1,-1),(0,1),(1,3),(4,-1)

The accompanying figure shows the graph of a cubic polynomial. Find the polynomial.

a. Find an equation that represents the family of all second degree polynomials that pass through the points \((0,1) \text { and }(1,2)\). [Hint: The equation will involve one arbitrary parameter that produces the members of the family when varied.] b. By hand, or with the help of a graphing utility, sketch four curves in the family. Equation Transcription: Text Transcription: (0,1) and (1,2)

In this section we have selected only a few applications of linear systems. Using the Internet as a search tool, try to find some more real-world applications of such systems. Select one that is of interest to you and write a paragraph about it.

In parts (a)–(e) determine whether the statement is true or false, and justify your answer. a. In any network, the sum of the flows out of a node must equal the sum of the flows into a node.

In parts (a)–(e) determine whether the statement is true or false, and justify your answer. b. When a current passes through a resistor, there is an increase in the electrical potential in a circuit.

In parts (a)–(e) determine whether the statement is true or false, and justify your answer. c. Kirchhoff’s current law states that the sum of the currents flowing into a node equals the sum of the currents flowing out of the node.

In parts (a)–(e) determine whether the statement is true or false, and justify your answer. d. A chemical equation is called balanced if the total number of atoms on each side of the equation is the same.

In parts (a)–(e) determine whether the statement is true or false, and justify your answer. e. Given any n points in the \(x y \text { - plane }\), there is a unique polynomial of degree \(n-1\) or less whose graph passes through those points. Equation Transcription: Text Transcription: xy-plane n-1

The following table shows the lifting force on an aircraft wing measured in a wind tunnel at various wind velocities. Model the data with an interpolating polynomial of degree 5, and use that polynomial to estimate the lifting force at \(2000 \mathrm{ft} / \mathrm{s}\). Equation Transcription: Text Transcription: 2000 ft/s

(Calculus required) Use the method of Example 7 to approximate the integral \(\int_{0}^{1} e^{x^{2}} d x\) by subdividing the interval of integration into five equal parts and using an interpolating polynomial to approximate the integrand. Compare your answer to that obtained using the numerical integration capability of your technology utility. Equation Transcription: Text Transcription: \int_0^1 e^x^2 d x

Use the method of Example 5 to balance the chemical equation. \(\mathrm{Fe}_{2} \mathrm{O}_{3}+\mathrm{Al} \rightarrow \mathrm{Al}_{2} \mathrm{O}_{3}+\mathrm{Fe}\) (Fe = iron, Al = aluminum, O = oxygen) Equation Transcription: Text Transcription: Fe_2 O_3 + Al \rightarrow Al_2 O_3 + F_e

Determine the currents in the accompanying circuit. Equation Transcription: Text Transcription: I_1 I_2 I_3 I_1 I_2 I_3 20 V 12 V 3 \omega 470 \omega 2 \omega