Exercises: Systems of non-linear equations

Solve the following system of non-linear equations:

$y = x^2 - 4x$

$y = x - 6$

Find the solution to the system:

$x + y = 7$

$x \times y = 12$

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

$x - y = 2$

$x^2 + y^2 = 34$

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

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

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-angled triangle measures $13 \text{ cm}$. It is known that one leg is $7 \text{ cm}$ longer than the other. Set up a system of non-linear equations to calculate the length of the two 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:

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

$x \times y = 18$

A group of students buys a present for a classmate for $2\,400$ pesos, sharing the cost equally. At the last minute, $2$ more students join, which means each of the original students pays $60$ pesos less. How many students were there at the beginning and how much did each 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 $192$. 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 piece of wire $40 \text{ cm}$ long is cut into two pieces. One of the pieces is used to form a square and the other to form a rectangle whose base is twice its height. If the sum of the areas of both shapes is $48 \text{ cm}^2$, how long is each piece of wire?

A number consists of two digits. The product of these digits is $18$. If $27$ units are subtracted from this number, another number with the same digits in reverse order is obtained. 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:

$x^2 + y^2 = 25$

$x \times y = 12$

A travel agency organizes a trip for a total cost of $12\,000$ pesos. Shortly before departure, $5$ people cancel their trip, forcing the others to pay $80$ pesos more each to cover the total cost. How many people actually went on the trip and how much did each pay?

In a park, a rectangular play area will be built surrounded by an artificial grass path of width $1 \text{ m}$ along 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 non-linear equations:

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

$x + y = 5$

A cyclist covers a distance of $120 \text{ km}$ at a constant speed. If they had travelled $5 \text{ km/h}$ 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 (curved surface area plus two bases) of $150\pi \text{ cm}^2$. Furthermore, it is 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 $A = 2\pi r^2 + 2\pi rh$).