1. Fundamental Concepts of Algebra

2. Equations and Inequalities

3. Functions and Graphs

4. Polynomial and Rational Functions

5. Exponential and Logarithmic Functions

6. Systems of Equations and Inequalities

7. Matrices and Determinants

8. Conic Sections

9. Sequences, Induction, and Probability

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Answer each question. A line passes through the points of intersection of the graphs of y = x^2 and x^2 + y^2 = 90. What is the equation of this line? <img src="https://lightcat-files.s3.amazonaws.com/problem_images/034d2cb00a5d7dd5-1685622727587.jpg" alt="" />

In Exercises 19–28, solve each system by the addition method. x^2+y^2=13,x^2−y^2=5

Non-Linear Systems

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
x^2 + 2y^2 = 9 
x^2 + y^2 = 25

In Exercises 1–18, solve each system by the substitution method. x+y=2, y=x^2−4

In Exercises 47–52, solve each system by the method of your choice. 2x^2+xy=6, x^2+2xy=0

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
3x^2 - y^2 = 11 
xy = 12

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
y = 6x + x^2 
4x - y = -3

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
x^2 + y^2 = 0 
2x^2 - 3y^2 = 0

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
y = x^2 + 6x + 9 
x + 2y = -2

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
5x^2 - 2y^2 = 25 
10x^2 + y^2 = 50

In Exercises 19–28, solve each system by the addition method. y^2−x=4, x^2+y^2=4

In Exercises 1–18, solve each system by the substitution method. xy=6, 2x-y=1

Solve each problem using a system of equations in two variables. See Example 6. Find two numbers whose sum is 17 and whose product is 42.

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
y = x^2 - 2x + 1 
x - 3y = -1

What are Linear and Nonlinear Equations?

In Exercises 29–42, solve each system by the method of your choice. x^2+4y^2=20, x+2y=6

Solve each problem using a system of equations in two variables. See Example 6. The longest side of a right triangle is 13 m in length. One of the other sides is 7 m longer than the shortest side. Find the lengths of the two shorter sides of the triangle.

Applications of systems of nonlinear equations

Solve each problem using a system of equations in two variables. See Example 6. Find two numbers whose squares have a sum of 100 and a difference of 28.

In Exercises 47–52, solve each system by the method of your choice. 3/x^2+1/y^2=7, 5/x^2−2/y^2=−3

In Exercises 29–42, solve each system by the method of your choice. x^2+y^2+3y=22, 2x+y=−1

Algebra 2 – Solving Linear-Nonlinear Systems

Solve systems of linear and quadratic equations using substitution

Solve each problem. The supply and demand equations for a certain commodity are given.
supply: p = 2000/(2000 - q) and demand: p = (7000 - 3q)/2q

Find the equilibrium price (in dollars).

In Exercises 25–35, solve each system by the method of your choice.  This  is a piecewise function, refer to textbook problem.

Solve each problem. Find the radius and height (to the nearest thousandth) of an open-ended cylinder with volume 50 in.^3 and lateral surface area 65 in.^2.

Solve each problem. The supply and demand equations for a certain commodity are given.
supply: p = √(0.1q + 9) - 2 and demand: p = √(25 - 0.1q)

Find the equilibrium demand.

In Exercises 19–28, solve each system by the addition method. 3x^2+4y^2−16=0, 2x^2−3y^2−5=0

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
x^2 - y = 0 
x + y = 2

In Exercises 19–28, solve each system by the addition method. x^2−4y^2=−7, 3x^2+y^2=31

In Exercises 29–42, solve each system by the method of your choice. 2x^2+y^2=18, xy=4

Solve each problem using a system of equations in two variables. See Example 6. Find two numbers whose ratio is 9 to 2 and whose product is 162.

Solve a system of a quadratic and linear equation alegbriacally (no real number solutions)

System of Nonlinear Equations -- Area and Perimeter Application

In Exercises 1–18, solve each system by the substitution method. x+y=1, x^2+xy-y^2=-5

In Exercises 29–42, solve each system by the method of your choice. x^3+y=0, x^2−y=0

Solve a system of a quadratic and linear equation algebrically (one solution)

In Exercises 1–18, solve each system by the substitution method. x+y=2, y=x^2-4x+4

In Exercises 1–18, solve each system by the substitution method. x^2+y^2=25, x-y=1

The perimeter of a rectangle is 26 meters and its area is 40 square meters. Find its dimensions.

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
2xy + 1 = 0 
x + 16y = 2

In Exercises 29–42, solve each system by the method of your choice. y=(x+3)^2, x+2y=−2

In Exercises 47–52, solve each system by the method of your choice. −4x+y=12, y=x^3+3x^2

Linear & Nonlinear Functions

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 nonlinear equations. Solve the system and find the numbers. The sum of two numbers is 10 and their product is 24. Find the numbers.

In Exercises 29–42, solve each system by the method of your choice. 3x^2+4y^2=16, 2x^2−3y^2=5

Solve the systems in Exercises 79–80. log_y x=3, log_y (4x)=5

Solve Problems Using Systems of Nonlinear Equations

Systems of Nonlinear Equations in Two Variables

Solving Applied Problems Using a System of Nonlinear Equations

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 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.

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
3x^2 + 5y^2 = 17 
2x^2 - 3y^2 = 5

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
x^2 + y^2 = 10 
2x^2 - y^2 = 17

Determining if equations are linear

Verify that the points of intersection specified on the graph of each nonlinear system are solutions of the system by substituting directly into both equations. 
2x^2 = 3y + 23
y = 2x - 5 <img src="https://lightcat-files.s3.amazonaws.com/problem_images/65aaf53da39d730e-1685625602154.jpg" alt="" />

Verify that the points of intersection specified on the graph of each nonlinear system are solutions of the system by substituting directly into both equations. 
y = 3x^2 
x^2 + y^2 = 10 <img src="https://lightcat-files.s3.amazonaws.com/problem_images/cba9025adf4a07c7-1682506353035.jpg" alt="" />

Answer each of the following. When appropriate, fill in the blank to correctly complete the sentence. The following nonlinear system has two solutions, one of which is (3,____). 
x + y = 7 
x^2 + y^2 = 25

In Exercises 19–28, solve each system by the addition method. x^2+y^2=25, (x−8)^2+y^2=41

Answer each of the following. When appropriate, fill in the blank to correctly complete the sentence. If we want to solve the following nonlinear system by substitution and we decide to solve equation (2) for y, what will be the resulting equation when the substitution is made into equation (1)? 
x^2 + y = 2 (1) 
x - y = 0 (2)

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
x^2 + y^2 = 5 
-3x + 4y = 2

Answer each of the following. When appropriate, fill in the blank to correctly complete the sentence. The following nonlinear system has two solutions, one of which is (___, 3).
2x + y = 1
x^2 + y^2 = 10

Solve the systems in Exercises 79–80. log x^2=y+3, log x=y−1

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

In Exercises 1–18, solve each system by the substitution method. y^2=x^2-9, 2y=x-3

Find the length and width of a rectangle whose perimeter is 40 feet and whose area is 96 square feet.

Solve each problem using a system of equations in two variables. See Example 6. Find two numbers whose ratio is 4 to 3 and are such that the sum of their squares is 100.

In Exercises 1–18, solve each system by the substitution method. x+y=1, (x-1)^2+(y+2)^2=10

Applications of Nonlinear Systems of Equations in Two Variables

Solve each problem. The supply and demand equations for a certain commodity are given.
supply: p = 2000/(2000 - q) and demand: p = (7000 - 3q)/2q

Find the equilibrium demand.

Solve each problem. The supply and demand equations for a certain commodity are given.
supply: p = √(0.1q + 9) - 2 and demand: p = √(25 - 0.1q)

Find the equilibrium price (in dollars).

Solve the system <img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/cab1fece12d00f2b-1678241302484.jpg" style="width: 134px;">

In Exercises 29–42, solve each system by the method of your choice. x^2+(y−2)^2=4, x^2−2y=0

Solve each problem. Find the equation of the line passing through the points of intersection of the graphs of x^2 + y^2 = 20 and x^2 - y = 0.

In Exercises 1–18, solve each system by the substitution method. xy=3, x^2+y^2=10

In Exercises 1–18, solve each system by the substitution method. y=x^2-4x-10, y=-x^2-2x+14

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