Equations and systems · Equations, inequalities and systems
Non-linear systems of equations
The perimeter of a rectangle is $28$ cm and its area is $48$ cm². Once again we have two unknowns and two equations, but now one of them contains a product of unknowns instead of a sum. It is no longer linear: it is a non-linear system, and it needs a slightly different strategy.
A linear equation is a line; an equation with $x^2$ (and no $y^2$) is a parabola. A line–parabola system can have two, one or no real solutions, depending on the discriminant of the quadratic equation that results from substituting.
Practice★☆☆Not started
Solve the following system of non-linear equations: $y = x^2 - 4x$ $y = x - 6$
Solve the following system of non-linear equations:
In rectangles, rhombuses and other shapes, the perimeter usually gives a linear equation, and the area (or the diagonal, using Pythagoras) gives a non-linear equation.
Practice★☆☆Not started
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 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…
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 non-linear system can have several mathematical solutions, but not all of them make sense in a real-life problem.
Practice★★☆Not started
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…
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 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.…
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?
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…
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.