Chapter 5

Let \(x\) represent one number and let \(y\) represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers. The sum of two numbers is 10 and their product is \(24 .\) Find the numbers.

Describe in general terms how to solve a system in three variables.

Perform each long division and write the partial fraction decomposition of the remainder term. $$\frac{x^{5}}{x^{2}-4 x+4}$$

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x+y>3 \\ x+y>-2 \end{array}\right.$$

Let \(x\) represent one number and let \(y\) represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers. The sum of two numbers is 20 and their product is \(96 .\) Find the numbers

Perform each long division and write the partial fraction decomposition of the remainder term. $$\frac{x^{4}-x^{2}+2}{x^{3}-x^{2}}$$

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} y \geq x^{2}-1 \\ x-y \geq-1 \end{array}\right.$$

Let \(x\) represent one number and let \(y\) represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers. The difference between the squares of two numbers is 3 . Twice the square of the first number increased by the square of the second number is 9. Find the numbers.

Technology Exercises Does your graphing utility have a feature that allows you to solve linear systems by entering coefficients and constant terms? If so, use this feature to verify the solutions to any five exercises that you worked by hand from Exercises \(5-16\)

Perform each long division and write the partial fraction decomposition of the remainder term. $$\frac{x^{4}+2 x^{3}-4 x^{2}+x-3}{x^{2}-x-2}$$

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} y \geq x^{2}-4 \\ x-y \geq 2 \end{array}\right.$$

Let \(x\) represent one number and let \(y\) represent the other number. Use the given conditions to write a system of nonlinear equations. Solve the system and find the numbers. The difference between the squares of two numbers is \(5 .\) Twice the square of the second number subtracted from three times the square of the first number is \(19 .\) Find the numbers.

Write the partial fraction decomposition of each rational expression. $$\frac{1}{x^{2}-c^{2}} \quad(c \neq 0)$$

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x^{2}+y^{2} \leq 16 \\ x+y>2 \end{array}\right.$$

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} 2 x^{2}+x y-6 \\ x^{2}+2 x y-0 \end{array}\right.$$

Write the partial fraction decomposition of each rational expression. $$\frac{a x+b}{x^{2}-c^{2}} \quad(c \neq 0)$$

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x^{2}+y^{2} \leq 4 \\ x+y>1 \end{array}\right.$$

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} 4 x^{2}+x y-30 \\ x^{2}+3 x y--9 \end{array}\right.$$

Make Sense? In Exercises \(48-51\), determine whether each statement makes sense or does not make sense, and explain your reasoning. A system of linear equations in three variables, \(x, y,\) and \(z\) cannot contain an equation in the form \(y=m x+b\)

Write the partial fraction decomposition of each rational expression. $$\frac{a x+b}{(x-c)^{2}} \quad(c \neq 0)$$

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x^{2}+y^{2}>1 \\ x^{2}+y^{2}<16 \end{array}\right.$$

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} -4 x+y-12 \\ y-x^{3}+3 x^{2} \end{array}\right.$$

Solve each system for \(x\) and \(y,\) expressing either value in terms of a or \(b,\) if necessary. Assume that \(a \neq 0\) and \(b \neq 0\). \(\left\\{\begin{array}{c}5 a x+4 y-17 \\ a x+7 y-22\end{array}\right.\)

Make Sense? In Exercises \(48-51\), determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm solving a three-variable system in which one of the given equations has a missing term, so it will not be necessary to use any of the original equations twice when I reduce the system to two equations in two variables.

Write the partial fraction decomposition of each rational expression. $$\frac{1}{x^{2}-a x-b x+a b} \quad(a \neq b)$$

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x^{2}+y^{2}>1 \\ x^{2}+y^{2}<9 \end{array}\right.$$

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} -9 x+y-45 \\ y-x^{3}+5 x^{2} \end{array}\right.$$

Solve each system for \(x\) and \(y,\) expressing either value in terms of a or \(b,\) if necessary. Assume that \(a \neq 0\) and \(b \neq 0\). \(\left\\{\begin{array}{l}4 a x+b y-3 \\ 6 a x+5 b y-8\end{array}\right.\)

Make Sense? In Exercises \(48-51\), determine whether each statement makes sense or does not make sense, and explain your reasoning. Because the percentage of the U.S. population that was foreign-born decreased from 1910 through 1970 and then increased after that, a quadratic function of the form \(f(x)=a x^{2}+b x+c,\) rather than a linear function of the form \(f(x)=m x+b,\) should be used to model the data.

Find the partial fraction decomposition for \(\frac{1}{x(x+1)}\) and use the result to find the following sum: $$\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\dots+\frac{1}{99 \cdot 100}$$

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} (x-1)^{2}+(y+1)^{2}<25 \\ (x-1)^{2}+(y+1)^{2} \geq 16 \end{array}\right.$$

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} \frac{3}{x^{2}}+\frac{1}{y^{2}}-7 \\ \frac{5}{x^{2}}-\frac{2}{y^{2}}--3 \end{array}\right.$$

Describe how the system $$ \left\\{\begin{array}{c} x+y-z-2 w=-8 \\ x-2 y+3 z+w=18 \\ 2 x+2 y+2 z-2 w=10 \\ 2 x+y-z+w=3 \end{array}\right. $$

Find the partial fraction decomposition for \(\frac{2}{x(x+2)}\) and use the result to find the following sum: $$\frac{2}{1 \cdot 3}+\frac{2}{3 \cdot 5}+\frac{2}{5 \cdot 7}+\dots+\frac{2}{99 \cdot 101}$$

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} (x+1)^{2}+(y-1)^{2}<16 \\ (x+1)^{2}+(y-1)^{2} \geq 4 \end{array}\right.$$

Solve each system by the method of your choice. $$\left\\{\begin{array}{l} \frac{2}{x^{2}}+\frac{1}{y^{2}}-11 \\ \frac{4}{x^{2}}-\frac{2}{y^{2}}--14 \end{array}\right.$$

A modernistic painting consists of triangles, rectangles, and pentagons, all drawn so as to not overlap or share sides. Within each rectangle are drawn 2 red roses and each pentagon contains 5 carnations. How many triangles, rectangles, and pentagons appear in the painting if the painting contains a total of 40 geometric figures, 153 sides of geometric figures, and 72 flowers?

Explain what is meant by the partial fraction decomposition of a rational expression.

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{aligned} x^{2}+y^{2} & \leq 1 \\ y-x^{2} &>0 \end{aligned}\right.$$

Make a rough sketch in a rectangular coordinate system of the graphs representing the equations in each system. The system, whose graphs are a line with positive slope and a parabola whose equation has a positive leading coefficient, has two solutions.

Explain how to find the partial fraction decomposition of a rational expression with distinct linear factors in the denominator.

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{aligned} x^{2}+y^{2} &<4 \\ y-x^{2} & \geq 0 \end{aligned}\right.$$

Make a rough sketch in a rectangular coordinate system of the graphs representing the equations in each system. The system, whose graphs are a line with negative slope and a parabola whose equation has a negative leading coefficient, has one solution.

Exercises \(55-57\) will help you prepare for the material covered in the next section. $$ \text { Subtract: } \frac{3}{x-4}-\frac{2}{x+2} $$

Explain how to find the partial fraction decomposition of a rational expression with a repeated linear factor in the denominator.

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x^{2}+y^{2}<16 \\ y \geq 2^{x} \end{array}\right.$$

A planet's orbit follows a path described by \(16 x^{2}+4 y^{2}-64\) A comet follows the parabolic path \(y-x^{2}-4 .\) Where might the comet intersect the orbiting planet?

Exercises \(55-57\) will help you prepare for the material covered in the next section. $$ \text { Add }: \frac{5 x-3}{x^{2}+1}+\frac{2 x}{\left(x^{2}+1\right)^{2}} $$

Explain how to find the partial fraction decomposition of a rational expression with a prime quadratic factor in the denominator.

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l} x^{2}+y^{2} \leq 16 \\ y<2^{x} \end{array}\right.$$