(b) If the price is $\$ 20,$will there be a shortage or a surplus? (c) Find the equilibrium price and the corresponding equilibrium quantity. Anurag K. Numerade Educator Problem 34 As background you need to have read the discussion on supply and demand preceding Example$6,$as well as Example 6 itself. In each exercise, assume that the price$p$is in dollars. Assume that the supply and demand functions for a commodity are as follows. Supply:$q=15 p,$Demand:$q=12,493-50 p,$for$p \geq 0$(a) If the price is set at$\$50,$ will there be a shortage or
a surplus of the commodity? What if the price is tripled?

Anurag K.

Problem 38

A certain alloy contains $10 \%$ tin and $30 \%$ copper. (The percentages are by weight.) How many pounds of tin and how many pounds of copper must be melted with 1000 lb of the given alloy to yield a new alloy containing $20 \%$ tin and $35 \%$ copper? Hint: Introduce variables for the weights of tin and copper to be added to the given alloy. Express the total weight of the new alloy in terms of these variables. The total weight of tin in the new alloy can be computed two ways, giving one equation. Computing the total weight of copper similarly gives a second equation.

Anurag K.

Problem 39

In this exercise you'll check the result of Example $5,$ first visually, then algebraically.
(a) Graph the parabola $y=x^{2}+\frac{5}{4} x-\frac{17}{4}$ in an appropriate viewing rectangle to see that it appears to pass through the two points $(-3,1)$ and $(1,-2)$
(b) Verify algebraically that the two points $(-3,1)$ and $(1,-2)$ indeed lie on the parabola $y=x^{2}+\frac{5}{4} x-\frac{17}{4}$

Anurag K.

Problem 40

Consider the following system from Example 7 :
\left\{\begin{aligned} x+y &=100 \\ -2 x+3 y &=0 \end{aligned}\right.
(a) Solve this system using the method of substitution. (As was stated in Example $7,$ you should obtain $x=60$ $y=40 .)$
(b) Solve the system using the addition-subtraction method.

Anurag K.

Problem 41

Find constants $a$ and $b$ so that $(8,-7)$ is the solution of the system
$$\left\{\begin{array}{l} a x+b y=10 \\ b x+a y=-5 \end{array}\right.$$

Anurag K.

Problem 42

(a) Sketch the triangular region in the first quadrant bounded by the lines $y=5 x, y=-3 x+6,$ and the $x$ -axis. One vertex of this triangle is the origin. Find the coordinates of the other two vertices. Then use your answers to compute the area of the triangle.
(b) More generally now, express the area of the shaded triangle in the following figure in terms of $m, M,$ and $b$ Then use your result to check the answer you obtained in part (a) for that area.
(Figure can't copy)

Anurag K.

Problem 43

Find $x$ and $y$ in terms of $a$ and $b$ :
$$\left\{\begin{array}{l} \frac{x}{a}+\frac{y}{b}=1 \\ \frac{x}{b}+\frac{y}{a}=1 \end{array}\right.$$
Does your solution impose any conditions on a and $b$ ?

Anurag K.

Problem 44

Solve the following system for $x$ and $y$ in terms of $a$ and $b$ where $a \neq b$
$$\left\{\begin{array}{l} a x+b y=1 / a \\ b^{2} x+a^{2} y=1 \end{array}\right.$$

Anurag K.

Problem 45

Solve the following system for $x$ and $y$ in terms of $a$ and $b$ where $a \neq b$
$$\left\{\begin{array}{l} a x+a^{2} y=1 \\ b x+b^{2} y=1 \end{array}\right.$$
Does your solution impose any additional conditions on a and $b$ ?

Check back soon!

Problem 46

Solve the following system for $s$ and $t$
$$\frac{-1}{1-2}$$
Hint: Make the substitutions $1 / s=x$ and $1 / t=y$ in order to obtain a system of two linear equations.

Anurag K.

Problem 47

Solve the following system for $s$ and $t$
2
$\left\{\begin{array}{c}\frac{1}{2 s}-\frac{1}{2 t}=-10 \\ \frac{2}{s}+\frac{3}{t}=5\end{array}\right.$
(Use the hint in Exercise $46 .$ )

Anurag K.

Problem 48

Consider the following system: $\left\{\begin{array}{l}2 x^{2}+2 y^{2}=55 \\ 4 x^{2}-8 y^{2}=k\end{array}\right.$
(a) Assuming $k=109,$ solve the system. Hint: The substitutions $u=x^{2}$ and $v=y^{2}$ will give you a linear system.
(b) Follow part (a) using: $k=110 ; k=111$
(c) Use a graphing utility to shed light on why the number of solutions is different for each of the values of $k$ considered in parts (a) and (b).

Anurag K.

Problem 49

Find all solutions of the given systems. For Exercises $51-55,$ use a calculator to round the final answers to two decimal places.
$$\left\{\begin{array}{l} \frac{2 w-1}{3}+\frac{z+2}{4}=4 \\ \frac{w+3}{2}-\frac{w-z}{3}=3 \end{array}\right.$$

Anurag K.

Problem 50

Find all solutions of the given systems. For Exercises $51-55,$ use a calculator to round the final answers to two decimal places.
$$\frac{x-y}{2}=\frac{x+y}{3}=1$$

Anurag K.

Problem 51

Find all solutions of the given systems. For Exercises $51-55,$ use a calculator to round the final answers to two decimal places.
\left\{\begin{aligned} 2 \ln x-5 \ln y &=11 \\ \ln x+\ln y &=-5 \end{aligned}\right.
Hint: Let $u=\ln x$ and $v=\ln y$

Anurag K.

Problem 52

Find all solutions of the given systems. For Exercises $51-55,$ use a calculator to round the final answers to two decimal places.
$$\left\{\begin{array}{l} 3 \ln x+\ln y=3 \\ 4 \ln x-6 \ln y=-7 \end{array}\right.$$
Hint: Let $u=\ln x$ and $v=\ln y$

Anurag K.

Problem 53

Find all solutions of the given systems. For Exercises $51-55,$ use a calculator to round the final answers to two decimal places.
\left\{\begin{aligned} e^{x}-3 e^{y} &=2 \\ 3 e^{x}+e^{y} &=16 \end{aligned}\right.
Hint: Let $u=e^{x}$ and $v=e^{y}$

Anurag K.

Problem 54

Find all solutions of the given systems. For Exercises $51-55,$ use a calculator to round the final answers to two decimal places.
\left\{\begin{aligned} e^{x}+2 e^{y} &=4 \\ \frac{1}{2} e^{x}-e^{y} &=0 \end{aligned}\right.
Hint: Let $u=e^{x}$ and $v=e^{y}$

Anurag K.

Problem 55

Find all solutions of the given systems. For Exercises $51-55,$ use a calculator to round the final answers to two decimal places.
$$\left\{\begin{array}{c} 4 \sqrt{x^{2}-3 x}-3 \sqrt{y^{2}+6 y}=-4 \\ \frac{1}{2}(\sqrt{x^{2}-3 x}+\sqrt{y^{2}+6 y})=3 \end{array}\right.$$

Anurag K.

Problem 56

The sum of two numbers is 64. Twice the larger number plus five times the smaller number is $20 .$ Find the two numbers. (Let $x$ denote the larger number and let $y$ denote the smaller number.)

Anurag K.

Problem 57

In a two-digit number, the sum of the digits is 14. Twice the tens digit exceeds the units digit by one. Find the number.

Anurag K.

Problem 58

You have two brands of dietary supplements on your shelf. Among other ingredients, both contain protein and carbohydrates. The amounts of protein and carbohydrates in one unit of each supplement are given in the following table as percentages of the recommended daily amount (RDA). How many units of each supplement do you need in a day to obtain the RDA for both protein and carbohydrates?
$$\begin{array}{lcc} & \begin{array}{c} \text { Protein } \\ \text { (\% of RDA } \\ \text { in one unit) } \end{array} & \begin{array}{c} \text { Carbohydrates } \\ \text { (\% of RDA } \\ \text { in one unit) } \end{array} \\ \hline \text { Supplement #1 } & 8 & 12 \\ \text { Supplement #2 } & 16 & 4 \\ \hline \end{array}$$

Anurag K.

Problem 59

Solve the following system for $x$ and $y$ in terms of $a$ and $b$ where $a$ and $b$ are nonzero and $a \neq b$
$$\left\{\begin{array}{l} \frac{a}{b x}+\frac{b}{a y}=a+b \\ \frac{b}{x}+\frac{a}{y}=a^{2}+b^{2} \end{array}\right.$$

Anurag K.

Problem 60

Solve for $x$ and $y$ in terms of $a, b, c, d, e,$ and $f$
$$\left\{\begin{array}{l} a x+b y=c \\ d x+e y=f \end{array}\right.$$
(Assume that $a e-b d \neq 0$.)

Anurag K.

Problem 61

(a) Given that the lines $7 x+5 y=4, x+k y=3,$ and $5 x+y+k=0$ are concurrent (pass through a common point), what are the possible values for $k ?$
(b) Check that your answers are reasonable: For each value of $k$ that you find, use a graphing utility to draw the three lines. Do they appear to be concurrent?

Anurag K.
Solve the following system for $x$ and $y$ in terms of $a$ and $b$ where $a b \neq-1:$
$$\left\{\begin{array}{l} \frac{x+y-1}{x-y+1}=a \\ \frac{y-x+1}{x-y+1}=a b \end{array}\right.$$