Chapter 5

In Exercises \(43-46,\) let \(x\) represent one number and let \(y\) represent the other number. Use the given conditions to write a system of equations. Solve the system and find the numbers. Three times a first number decreased by a second number is 1. The first number increased by twice the second number is \(12 .\) Find the numbers.

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

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.

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} y \geq x^{2}-4 \\ x-y \geq 2 \end{array}\right. $$

In Exercises \(43-46,\) let \(x\) represent one number and let \(y\) represent the other number. Use the given conditions to write a system of equations. Solve the system and find the numbers. The sum of three times a first number and twice a second number is \(8 .\) If the second number is subtracted from twice the first number, the result is \(3 .\) Find the numbers.

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

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} x^{2}+y^{2} \leq 16 \\ x+y>2 \end{array}\right. $$

In Exercises \(47-48,\) solve each system by the method of your choice. $$ \left\\{\begin{array}{l} \frac{x+2}{2}-\frac{y+4}{3}=3 \\ \frac{x+y}{5}=\frac{x-y}{2}-\frac{5}{2} \end{array}\right. $$

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

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

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

In Exercises \(47-48,\) solve each system by the method of your choice. $$ \left\\{\begin{array}{l} \frac{x-y}{3}=\frac{x+y}{2}-\frac{1}{2} \\ \frac{x+2}{2}-4=\frac{y+4}{3} \end{array}\right. $$

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. $$ \frac{a x+b}{(x-c)^{2}} \quad(c \neq 0) $$

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

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

In Exercises \(49-50,\) solve each system for \(x\) and \(y,\) expressing either value in terms of a or \(b,\) if necessary. Assume that \(a \neq 0\) and \(b \neq 0\) $$ \left\\{\begin{aligned} 5 a x+4 y &=17 \\ a x+7 y &=22 \end{aligned}\right. $$

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. $$ \frac{1}{x^{2}-a x-b x+a b} \quad(a \neq b) $$

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

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

In Exercises \(49-50,\) solve each system for \(x\) and \(y,\) expressing either value in terms of a or \(b,\) if necessary. Assume that \(a \neq 0\) and \(b \neq 0\) $$ \left\\{\begin{array}{l} 4 a x+b y=3 \\ 6 a x+5 b y=8 \end{array}\right. $$

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: $$ \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\dots+\frac{1}{99 \cdot 100} $$

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

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

For the linear function \(f(x)=m x+b, f(-2)=11\) and \(f(3)=-9 .\) Find \(m\) and \(b\)

Describe how the system $$\left\\{\begin{aligned} x+y-z-2 w &=-8 \\ x-2 y+3 z+w &=18 \\ 2 x+2 y+2 z-2 w &=10 \\ 2 x+y-z+w &=3 \end{aligned}\right.$$ could be solved. Is it likely that in the near future a graphing utility will be available to provide a geometric solution (using Fintersecting graphs) to this system? Explain.

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

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

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

For the linear function \(f(x)=m x+b, f(-3)=23\) and \(f(2)=-7 .\) Find \(m\) and \(b\)

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.

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.

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

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

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.

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

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.

Application Exercises 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?

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l}x^{2}+y^{2}<16 \\\y \geq 2^{x}\end{array}\right.$$

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.

Application Exercises A system for tracking ships indicates that a ship lies on a path described by \(2 y^{2}-x^{2}=1 .\) The process is repeated and the ship is found to lie on a path described by \(2 x^{2}-y^{2}=1 .\) If it is known that the ship is located in the first quadrant of the coordinate system, determine its exact location.

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l}x^{2}+y^{2} \leq 16 \\\y<2^{x}\end{array}\right.$$

Solve: $$\left\\{\begin{array}{r} A+B=3 \\ 2 A-2 B+C=17 \\ 4 A-2 C=14 \end{array}\right.$$

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

Application Exercises Find the length and width of a rectangle whose perimeter is 36 feet and whose area is 77 square feet.