Systems of Equations and Inequalities

Graph the equations given in Example $4$.

Graph the equations given in Problem $49$.

In Problems $57-64$, use a graphing utility to solve each system of equations. Express the solution(s) rounded to two decimal places.

$\left\{\begin{array}{l}y=x^{\frac{2}{3}}\\ y=e^{-x}\end{array}\right.$

In Problems $57-64$, use a graphing utility to solve each system of equations. Express the solution(s) rounded to two decimal places.

$\left\{\begin{array}{l}y=x^{\frac{3}{2}}\\ y=e^{-x}\end{array}\right.$

In Problems $57-64$, use a graphing utility to solve each system of equations. Express the solution(s) rounded to two decimal places.

$\left\{\begin{array}{l}{x}^2+y^3=2\\ x^{3}y=4\end{array}\right.$

In Problems $57-64$, use a graphing utility to solve each system of equations. Express the solution(s) rounded to two decimal places.

$\left\{\begin{array}{l}{x}^3+y^2=2\\ x^{2}y=4\end{array}\right.$

In Problems $57-64$, use a graphing utility to solve each system of equations. Express the solution(s) rounded to two decimal places.

$\left\{\begin{array}{l}{x}^4+y^4=12\\ x{y}^2=2\end{array}\right.$

In Problems $57-64$, use a graphing utility to solve each system of equations. Express the solution(s) rounded to two decimal places.

$\left\{\begin{array}{l}{x}^4+y^4=6\\ xy=1\end{array}\right.$

In Problems $57-64$, use a graphing utility to solve each system of equations. Express the solution(s) rounded to two decimal places.

$\left\{\begin{array}{l}xy=2\\ y=\ln x\end{array}\right.$

In Problems $57-64$, use a graphing utility to solve each system of equations. Express the solution(s) rounded to two decimal places.

$\left\{\begin{array}{l}{x}^2+y^2=4\\ y=\ln x\end{array}\right.$

In Problems $65-70$, graph each equation and find the point(s) of intersection, if any.

The line $x+2y=0$ and the circle ${\left(x-1\right)}^2+{\left(y-1\right)}^2=5$.

In Problems $65-70$, graph each equation and find the point(s) of intersection, if any.

The line $x+2y+6=0$ and the circle ${\left(x+1\right)}^2+{\left(y+1\right)}^2=5$. 

In Problems $65-70$, graph each equation and find the point(s) of intersection, if any.

The circle ${\left(x-1\right)}^2+{\left(y+2\right)}^2=4$ and the parabola $y^2+4y-x+1=0$.

In Problems $65-70$, graph each equation and find the point(s) of intersection, if any.

The circle ${\left(x+2\right)}^2+{\left(y-1\right)}^2=4$ and the parabola $y^2-2y-x-5=0$.

In Problems $65-70$, graph each equation and find the point(s) of intersection, if any.

$y=\frac{4}{x-3}$ and the circle $x^2-6x+y^2+1=0$.

In Problems $65-70$, graph each equation and find the point(s) of intersection, if any.

$y=\frac{4}{x+2}$ and the circle $x^2+4x+y^2-4=0$.

Solve the system. Use any method you wish.

$\left\{\begin{array}{l}{y}^2+y+x^2-x-2=0\\ y+1+\frac{x-2}{y}=0\end{array}\right.$

Solve the system. Use any method you wish.

$\left\{\begin{array}{l}{x}^3-2x^2+y^2+3y-4=0\\ x-2+\frac{y^2-y}{x^2}=0\end{array}\right.$

Solve the system. Use any method you wish.

$\left\{\begin{array}{l}{\log }_{x}y=3\\ {\log }_x\left(4y\right)=5\end{array}\right.$

Solve the system. Use any method you wish.

$\left\{\begin{array}{l}{\log }_x\left(2y\right)=3\\ {\log }_x\left(4y\right)=2\end{array}\right.$

The difference of two numbers is $2$, and the sum of their

squares is $10$. Find the numbers.

The sum of two numbers is $7$, and the difference of their

squares is $21$. Find the numbers.

The product of two numbers is $4$, and the sum of their squares is $8$. Find the numbers.

The product of two numbers is $10$, and the difference of their squares is $21$. Find the numbers.

The difference of two numbers is the same as their product, and the sum of their reciprocals is $5$. Find the numbers.

The sum of two numbers is the same as their product, and the difference of their reciprocals is $3$. Find the numbers.

The ratio of a to b is $\frac{2}{3}$. The sum of a and b is $10$. What is the ratio of $a + b$ to $b - a$? 

The ratio of a to b is $4:3$. The sum of a and b is $14$. What is the ratio of $a-b$ to $a+b$? 

The perimeter of a rectangle is $16$inches, and its area is $15 inche{s}^2$. What are its dimensions?

An area of $52 fee{t}^2$ is to be enclosed by two squares whose sides are in the ratio of $2:3$. Find the sides of the squares.

Two circles have circumferences that add up to $12\mathrm{\pi } \mathrm{cm}$ and areas that add up to $20\mathrm{\pi } {\mathrm{cm}}^2$. Find the radius of each circle.

The altitude of an isosceles triangle drawn to its base is $3 cm$, and its perimeter is $18 cm$. Find the length of its base.

The Tortoise and the Hare: In a 21-meter race between a tortoise and a hare, the tortoise leaves 9 minutes before the hare. The hare, by running at an average speed of 0.5 meter per hour faster than the tortoise, crosses the finish line 3 minutes before the tortoise. What are the average speeds of the tortoise and the hare?

Running a Race : In a 1-mile race, the winner crosses the finish line 10 feet ahead of the second-place runner and 20 feet ahead of the third-place runner. Assuming that each runner maintains a constant speed throughout the race, by how many feet does the second-place runner beat the third-place runner?

Constructing a Box: A rectangular piece of cardboard, whose area is 216 square centimeters, is made into an open box by cutting a 2-centimeter square from each corner and turning up the sides. See the figure. If the box is to have a volume of 224 cubic centimeters, what size cardboard should you start with?

Constructing a Cylindrical Tube: A rectangular piece of cardboard, whose area is 216 square centimeters, is made into a cylindrical tube by joining together two sides of the rectangle. See the figure. If the tube is to have a volume of 224 cubic centimeters, what size cardboard should you start with?

Fencing: A farmer has 300 feet of fence available to enclose a 4500-square-foot region in the shape of adjoining squares, with sides of length x and y. See the figure. Find x and y.

Bending Wire: A wire 60 feet long is cut into two pieces. Is it possible to bend one piece into the shape of a square and the other into the shape of a circle so that the total area enclosed by the two pieces is 100 square feet? If this is possible, find the length of the side of the square and the radius of the circle.

Geometry: Find formulas for the length l and width w of a rectangle in terms of its area A and perimeter P.

Geometry: Find formulas for the base b and one of the equal sides l of an isosceles triangle in terms of its altitude h and perimeter P.

Descartes’s Method of Equal Roots: Descartes’s method for finding tangents depends on the idea that, for many graphs, the tangent line at a given point is the unique line that intersects the graph at that point only. We apply his method to find an equation of the tangent line to the parabola $y=x^2$ at the point (2,4). See the figure.

First, we know that the equation of the tangent line must be in the form $y = mx + b$. Using the fact that the point $(2,4)$ is on the line, we can solve for b in terms of m and get the equation $y = mx + (4 - 2m)$. Now we want $(2,4)$ to be the unique solution to the system

$$\begin{gathered} y=x^2 \\ y=mx+4-2m \end{gathered}$$

From this system, we get $x^2-mx +(2m - 4)= 0$. By using the quadratic formula, we get

$x=\frac{m\pm \sqrt{m^2-4(2m-4)}}{2}$

To obtain a unique solution for x, the two roots must be equal; in other words, the discriminant $m^2-4(2m-4)$ must be 0.

Complete the work to get m, and write an equation of the tangent line.

In Problems 92–98, use Descartes’s method from Problem 91 to find the equation of the line tangent to each graph at the given point.

92. $x^2+y^2=10;$ at $(1,3)$.

In Problems 92–98, use Descartes’s method from Problem 91 to find the equation of the line tangent to each graph at the given point.

93. $y=x^2+2$ ; at $(1,3)$.

In Problems 92–98, use Descartes’s method from Problem 91 to find the equation of the line tangent to each graph at the given point.

94. $x^2+y=5$; at $(-2,1)$.

Use Descartes’s method  to find the equation of the line tangent to the graph $2x^2+3y^2=14$ at the given point (1,2).  

Use Descartes’s method  to find the equation of the line tangent to the graph $3x^2+y^2=7$ at the given point (-1,2). 

Use Descartes’s method  to find the equation of the line tangent to the graph  $x^2-y^2=3$at the given point (2,1). 

 Use Descartes’s method  to find the equation of the line tangent to the graph $2y^2-x^2=14$ at the given point (2,3).

If r1 and r2 are two solutions of a quadratic equation ax2 + bx + c = 0, it can be shown that

$r_1+r_2=-\frac{b}{a} and r_1{r}_2=\frac{c}{a}$

Solve this system of equations for r1 and r2.

A circle and a line intersect at most twice. A circle and a parabola intersect at most four times. Deduce that a circle and the graph of a polynomial of degree 3 intersect at most six times. What do you conjecture about a polynomial of degree 4? What about a polynomial of degree n? Can you explain your conclusions using an algebraic argument?