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:

$x + y = 7$

$x \cdot y = 12$

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:

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

$x^2 - y^2 = 32$

Sara tries to solve the following system of non-linear equations, which describes two numbers whose difference is $3$ and the sum of their squares is $29$:

To do this, she isolates $x = y + 3$ in the first equation and substitutes it into the second, writing in her notebook:

And she concludes that the only solution is $y = \sqrt{10}$, $x = 3 + \sqrt{10}$.

1. a) Identify the two serious conceptual errors Sara made in her working and explain in detail why they are incorrect.
2. b) Solve the system correctly, showing all algebraic steps, and state all valid pairs of solutions $(x, y)$.

We want to design a system of non-linear equations consisting of a vertical parabola of the form $y = x^2 + b$ and a straight line of the form $y = m\cdot x + n$.

1. a) Find the values of the parameters $b$, $m$, and $n$ so that the only solutions to the system are exactly the points $(0, 2)$ and $(3, 11)$. Justify how you deduce each value from the given coordinates.
2. b) Write down the resulting complete system and verify algebraically, solving it by equating the expressions, that no other intersection point exists between the two curves.

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:

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

$x \cdot y = 18$

A group of students buys a present for a classmate for $120$ euros, sharing the cost equally. At the last minute, $2$ more students join, which means each of the original students pays $3$ euros 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.

In a secondary school in the Valencian Community, a rectangular school garden is to be fenced off using an existing $20\text{ m}$ long wall, so that this wall serves as one of the sides and does not need fencing. With an allocated budget, exactly $28\text{ m}$ of wire mesh have been purchased to fence the other three sides (two sides perpendicular to the wall, of length $x$, and one side parallel to the wall, of length $y$). The Biology and Geology department requires the total cultivation area to be $96\text{ m}^2$.

1. a) Before setting up the system, make a reasoned estimate: knowing that side $y$ cannot measure more than the $20\text{ m}$ of the available wall, is it possible for side $x$ to measure less than $4\text{ m}$ or more than $11\text{ m}$? Justify your answer by analysing the available fencing and the required area together.
2. b) Set up the system of non-linear equations (perimeter and area) and solve it by substitution.
3. c) Check whether the mathematical solutions obtained are consistent with your initial estimate and determine the feasible dimensions the garden can have, taking into account the physical limitation of the wall.

To solve the following symmetric system of non-linear equations:

two Year 10 students propose different approaches:

• Laura's method (Substitution): Isolate $y = 15 - x$ in the first equation, substitute into the second, expand the square of the binomial, and solve the quadratic equation in $x$.

• Marcos's method (Notable identities): Square both sides of the first equation to obtain $(x + y)^2 = 225$, apply the expansion $(x + y)^2 = x^2 + y^2 + 2\cdot x\cdot y$, substitute $x^2 + y^2 = 117$ to directly calculate the product $x\cdot y$, and, using the sum $S$ and the product $P$, set up the equation $t^2 - S\cdot t + P = 0$.

1. a) Solve the system following Laura's method in detail.
2. b) Solve the system following Marcos's method in detail.
3. c) Compare both methods: which of the two paths requires simpler arithmetic calculations and presents a lower risk of making sign errors? Justify which one you would choose if the numbers in the problem were much larger.

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:

$x^2 + y^2 = 25$

$x \cdot y = 12$

A travel agency organises a trip for a total cost of $600$ euros. Shortly before departure, $5$ people cancel their trip, forcing the others to pay $4$ euros 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:

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

$x + y = 5$

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 $A = 2\pi r^2 + 2\pi rh$).

An agricultural cooperative in Castilla-La Mancha plans the installation of a rectangular structure for solar panels intended for the self-consumption of its well pumps. The technical specifications impose three design conditions:

1. For reasons of structural strength and the anchoring of diagonal cables, the diagonal of the rectangle must measure exactly $17\text{ m}$.

2. To guarantee the peak power required by the motors, the solar collection area must be exactly $120\text{ m}^2$.

3. To avoid shadows cast between adjacent rows of panels according to the solar radiation tables in the area, the length (longer side, $x$) must exceed the width (shorter side, $y$) by at least $6\text{ m}$.

1. a) Set up a system of non-linear equations relating the dimensions $x$ and $y$ using the diagonal and the area.
2. b) Solve the system by applying notable identities to calculate $(x+y)^2$ and $(x-y)^2$, deducing all possible mathematical solutions for the dimensions.
3. c) Interpret the results obtained in the context of the project and determine whether the structure's dimensions are feasible under the technical restriction of minimum separation to prevent shading. Justify your conclusion.

In a drone positioning and guidance system, a surveillance drone follows a parabolic trajectory given by the equation $y = x^2 - 4\cdot x + 7$, while a beacon light beam is emitted in a straight line following the equation $y = 2\cdot x + k$, where $k$ is a real parameter that can be adjusted by changing the tilt of the emitter.

1. a) Set up the system of equations formed by both trajectories and equate the expressions to obtain a quadratic equation of the form $a\cdot x^2 + b\cdot x + c = 0$ depending on $k$.
2. b) Without testing random values for $k$, use the discriminant ($\Delta = b^2 - 4\cdot a\cdot c$) to determine analytically what condition $k$ must satisfy so that:

• The light beam intersects the drone's trajectory at two distinct points.

• The light beam never reaches the drone's trajectory.

• The light beam is tangent to the drone's trajectory (a single point of contact).

1. c) For the value of $k$ that produces tangency, calculate the exact coordinates of the point of contact $(x, y)$ and explain the geometric meaning of the associated quadratic equation having a double root.