A rectangular rug for the living room has a perimeter of $28$ m and an area of $48$ m². How long are its sides? There are two unknowns and two equations, but now one of them contains a product of unknowns. It is no longer a linear system, and it needs a somewhat different strategy.
What a nonlinear system is
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Solve the system formed by a linear equation and a quadratic equation: $x - y = 2$ $x^2 + y^2 = 34$
Solve the system formed by a linear equation and a quadratic equation:
A linear equation is a line; an equation like $y = x^2 - 4x$ is a parabola. Solving the system means finding where they cross. When you substitute, you get a quadratic equation, and depending on its discriminant the line and the parabola have two, one, or no intersection points.
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Solve the following system of non-linear equations: $y = x^2 - 4x$ $y = x - 6$
Solve the following system of non-linear equations:
When an equation contains $\sqrt{x}$, the trick is to isolate the root and square both sides to get rid of it. But squaring can let in solutions that are not solutions of the original system, so you have to check at the end.
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Solve the following system containing a radical equation: $x + y = 10$ $\sqrt{x} - y = -4$
Solve the following system containing a radical equation:
In rectangles, rhombuses, and other figures, the perimeter usually gives a linear equation and the area (or the diagonal, using the Pythagorean theorem) gives a nonlinear equation.
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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…
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.
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…
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.
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…
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 $A = 2\pi r^2 + 2\pi rh$).
A nonlinear system can have several mathematical solutions, but not all of them make sense in a real problem.
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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.…
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.
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…
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?
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…
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.
With the ideas above you can solve almost any problem: choose the unknowns, translate each piece of data into an equation, solve for a variable in the simplest one, substitute, and check the context.
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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.
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.
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…
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 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…
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 $10x + y$).