Use Gaussian elimination to balance the following chemical reactions. a. aCl2 + bKOH _

(from Henry Burchard Fines A College Algebra, 1905) A and B are alloys of silver and copper. An alloy that is 5 parts A and 3 parts B is 52% silver. One that is 5 parts A and 11 parts B is 42% silver. What are the percentages of silver in A and B, respectively?

(from Henry Burchard Fines A College Algebra, 1905) Two vessels, A and B, contain mixtures of alcohol and water. A mixture of 3 parts from A and 2 parts from B will contain 40% of alcohol; and a mixture of 1 part from A and 2 parts from B will contain 32% of alcohol. What are the percentages of alcohol in A and B, respectively?

(from Henry Burchard Fines A College Algebra, 1905) Two points move at constant rates along the circumference of a circle whose length is 150 ft. When they move in opposite senses they meet every 5 seconds; when they move in the same sense they are together every 25 seconds. What are their rates?

A grocer mixes dark roast and light roast coffee beans to sell what she calls a French blend and a Viennese blend. For French blend she uses a mixture that is 3 parts dark and 1 part light roast; for Viennese, she uses a mixture that is 1 part dark and 1 part light roast. If she has at hand 20 pounds of dark roast and 17 pounds of light roast, how many pounds each of French and Viennese blend can she make so as to have no waste?

Find the parabola \(y = ax^2 + bx + c\) passing through the points (?1, 9), (1, ?1), and (2, 3).

Find the parabola \(y = ax^2 + bx + c\) passing through the points (?2, ?6), (1, 6), and (3, 4).

Let \(P_i=\left(x_i, y_i\right) \in \mathbb{R}^2, i=1,2,3\). Assume \(x_1, x_2\), and \(x_3\) are distinct (i.e., no two are equal). a. Show that the matrix \(\left[\begin{array}{lll}1 & x_1 & x_1^2 \\1 & x_2 & x_2^2 \\1 & x_3 & x_3^2\end{array}\right]\) is nonsingular. \({ }^{18}\) ___________ \({ }^{18}\) No confusion intended here: \(x_i^2\) means \(\left(x_i\right)^2\), i.c., the square of the real number \(x_i\). b. Show that the system of equations \(\left[\begin{array}{lll}x_1^2 & x_1 & 1 \\x_2^2 & x_2 & 1 \\x_3^2 & x_3 & 1\end{array}\right]\left[\begin{array}{l}a \\b \\c\end{array}\right]=\left[\begin{array}{l}y_1 \\y_2 \\y_3\end{array}\right]\) always has a unique solution. (Hint: Try reordering a, b, and c.) Remark. If a = 0, then the points \(P_1, P_2\), and \(P_3\) lie on the line y = b x + c; thus, we have shown that three noncollinear points lie on a unique parabola \(y=a x^2+b x+c\).

Find the cubic \(y = ax^3 + bx^2 + cx + d\) passing through the points (?2, 5), (?1, ?3), (1, ?1), and (2, ?3).

A circle C passes through the points (2, 6), (?1, 7), and (?4, ?2). Find the center and radius of C. (Hint: The equation of a circle can be written in the form \(x^2 + y^2 + ax + by + c = 0\). Why?)

A circle C passes through the points (-7, -2), (-1, 4), and (1, 2). Find the center and radius of C.

Let \(P_i=\left(x_i, y_i\right) \in \mathbb{R}^2, i=1,2,3\). Let \(A=\left[\begin{array}{lll}x_1 & y_1 & 1 \\x_2 & y_2 & 1 \\x_3 & y_3 & 1\end{array}\right]\) . a. Show that the three points \(P_1, P_2\), and \(P_3\) are collinear if and only if the equation \(A \mathbf{x}=\mathbf{0}\) has a nontrivial solution. (Hint: A general line in \(\mathbb{R}^2\) is of the form ax + by + c = 0, where a and b are not both 0 .) b. Deduce that if the three given points are not collinear, then there is a unique circle passing through them. (Hint: If you set up a system of linear equations as suggested by the hint for Exercise 9 , you should use part a to deduce that the appropriate coefficient matrix is nonsingular.)

Use Gaussian elimination to balance the following chemical reactions. *a. \(a \mathrm{Cl}_2+b \mathrm{KOH} \leftrightharpoons c \mathrm{KCl}+d \mathrm{KClO}_3+e \mathrm{H}_2 \mathrm{O}\) b. \(a \mathrm{~Pb}\left(\mathrm{~N}_3\right)_2+b \mathrm{Cr}\left(\mathrm{MnO}_4\right)_2 \leftrightharpoons c \mathrm{Cr}_2 \mathrm{O}_3+d \mathrm{MnO}_2+e \mathrm{~Pb}_3 \mathrm{O}_4+f \mathrm{~N}_2\)

Use Gaussian elimination to solve for the following partial fraction decompositions. a. 4x3 7x x4 5x2 + 4 = A x 1 + B x + 1 + C x 2 + D x + 2 b. x (x + 1)(x2 + 1) = A x + 1 + Bx + C x2 + 1

In each of the circuits pictured in Figure 6.4, calculate the current in each of the wires. (_ is the standard abbreviation for ohms.)

Let A be the matrix given in Example 8. a. Show that we can find vectors \(\mathbf{x} \in \mathbb{R}^2\) satisfying \(A \mathbf{x}=\mathbf{x}\) by solving \(B \mathbf{x}=\mathbf{0}\), where \(B=\left[\begin{array}{rr}-0.40 & 0.45 \\0.40 & -0.45\end{array}\right]\) (See Exercise 1.4.5.) Give the general solution of \(B \mathbf{x}=\mathbf{0}\) in standard form. b. Find the solution \(\mathbf{x}\) of \(B \mathbf{x}=\mathbf{0}\) with \(x_1+x_2=1\). (Note that in our discussion of Example 8, we always had \(p_k+q_k=1\).) c. Compare your answer to part b with the vector \(\mathbf{x}_{\infty}\) obtained in Example 8.

Investigate (with a computer or programmable calculator) the cat/mouse population behavior in Example 6, choosing a variety of beginning populations, if a. ck+1 = 0.7ck + 0.1mk mk+1 = 0.2ck + mk b. ck+1 = 1.3ck + 0.2mk mk+1 = 0.1ck + mk c. ck+1 = 1.1ck + 0.3mk mk+1 = 0.1ck + 0.9mk

Suppose a living organism that can live to a maximum age of 3 years has Leslie matrix \(A=\left[\begin{array}{ccc}0 & 0 & 8 \\\frac{1}{2} & 0 & 0 \\0 & \frac{1}{4} & 0\end{array}\right]\) . Find a stable age distribution vector \(\mathbf{x}\), i.e., a vector \(\mathbf{x} \in \mathbb{R}^3\) with \(A \mathbf{x}=\mathbf{x}\).

In Example 9 , we took our initial vector to be \(\mathbf{x}_0=\left[\begin{array}{l}1 \\ 1\end{array}\right]\). a. Find the first ten terms of the sequence obtained by starting instead with \(\mathbf{x}_0^{\prime}=\left[\begin{array}{l}2 \\ 1\end{array}\right]\). b. Describe the sequence obtained by starting instead with \(\mathbf{x}_0^{\prime}=\left[\begin{array}{l}b \\ c\end{array}\right]\). (Hint: Use the fact that \(\left[\begin{array}{l}b \\ c\end{array}\right]=b\left[\begin{array}{l}1 \\ 0\end{array}\right]+c\left[\begin{array}{l}0 \\ 1\end{array}\right]\).)

(A computer or calculator may be helpful in solving this problem.) Find numbers a, b, c, d, e, and f so that the five points (0,2),(-3,0),(1,5),(1,1), and (-1,1) all lie on the conic \(a x^2+b x y+c y^2+d x+e y+f=0\) . Show, moreover, that a, b, c, d, e, and f are uniquely determined up to a common factor.