Exercises: Non-linear systems of equations

Solve the following system of nonlinear equations:

$y = x^2 - 4x$

$y = x - 6$

Find the solution to the system:

$x + y = 7$

$x \cdot y = 12$

Solve the system formed by a linear and a quadratic equation:

$x - y = 2$

$x^2 + y^2 = 34$

The product of two positive integers is $156$ and one of them is consecutive to the other. Set up a system of equations and find these numbers.

The area of a rectangular carpet is $6 \text{ m}^2$ and its perimeter measures $10 \text{ m}$. Set up a system of nonlinear equations and calculate the dimensions of the carpet.

Solve the following system:

$x^2 - y^2 = 16$

$x + y = 8$

The sum of the squares of two numbers is $25$, and one of them is $1$ unit less than the other. Set up a system of equations and find the numbers.

Find the solution to the system:

$\frac{x}{y} = 3$

$x^2 - y^2 = 32$

The hypotenuse of a right triangle measures $13 \text{ cm}$. It is known that one of the legs measures $7 \text{ cm}$ more than the other. Set up a system of nonlinear equations to calculate the length of both legs.

The sum of two numbers is $15$ and the sum of their reciprocals is $\frac{5}{18}$. Set up a system of equations and find the numbers.

Solve the following system of equations:

$\frac{x}{2} + \frac{y}{3} = 4$

$x \cdot y = 18$

A group of students buys a gift for a classmate for $120$ dollars, to be paid in equal parts. At the last minute, $2$ more students join, which means that each of the original students pays $3$ dollars less. How many students were there at the beginning and how much did each one pay?

A rectangular plot has an area of $600 \text{ m}^2$. If its length is reduced by $5 \text{ m}$ and its width is increased by $5 \text{ m}$, the area remains the same. Calculate the original dimensions of the plot.

Solve the following system containing a radical equation:

$x + y = 10$

$\sqrt{x} - y = -4$

The product of the ages of two siblings is $96$. In $4$ years, the product of their ages will be $168$. What are their current ages?

The area of a rhombus is $24 \text{ cm}^2$ and the sum of the lengths of its diagonals is $14 \text{ cm}$. Determine the length of both diagonals by setting up a system of equations.

A wire of length $40 \text{ cm}$ is cut into two pieces. One of the pieces is used to form a square and the other is used to form a rectangle whose base is twice its height. If the sum of the areas of both shapes is $43 \text{ cm}^2$, how long is each piece of wire?

A number consists of two digits. The product of these digits is $18$. If $27$ is subtracted from this number, another number is obtained with the same digits but in reverse order. What is the original number? (Remember that a two-digit number $xy$ is expressed algebraically as $10x + y$).

Solve the following symmetric system of equations:

$x^2 + y^2 = 25$

$x \cdot y = 12$

A travel agency organizes a trip for a total cost of $600$ dollars. Shortly before leaving, $5$ people cancel their trip, which forces the rest to pay $4$ dollars more each to cover the total cost. How many people finally went on the trip and how much did each person pay?

A rectangular play area is going to be built in a park, surrounded by an artificial grass path $1 \text{ m}$ wide around its entire perimeter. The area of the play area must be $48 \text{ m}^2$ and the total area (play area plus path) must be $96 \text{ m}^2$. Set up a system of equations to find the dimensions of the play area.

Solve the following system of nonlinear equations:

$x^2 - x \cdot y + y^2 = 7$

$x + y = 5$

A cyclist travels a distance of $120 \text{ km}$ at a constant speed. If they had traveled $5 \text{ km/h}$ faster, they would have taken $2$ hours less to complete the same journey. Set up a system of equations to calculate the cyclist's speed and the time taken.

A closed cylindrical container has a total surface area (lateral area plus the two bases) of $150\pi \text{ cm}^2$. It is also known that the sum of the base radius and the height of the cylinder is $15 \text{ cm}$. Calculate the radius and height of the container. (Remember that the total surface area of a cylinder is $A = 2\pi r^2 + 2\pi rh$).