ESO Y4A · Mathematics
Equations · Systems of equations
Non-linear systems
What a non-linear system is Line and parabola: points of intersection Sum and product of two numbers Geometry problems: perimeter and area Checking the solutions in context Summary
The perimeter of a rectangle is 28 cm and its area is 48 cm². Two clues, two unknowns ($x$ and $y$, the sides) once again... but now one of the equations contains a product of unknowns, not a sum. It is no longer a linear system: it is a non-linear system .
What a non-linear system is
It is a system in which at least one of the equations is not of first degree: it contains $x^2$, $x \cdot y$, $\sqrt{x}$ or similar expressions. The general strategy is still the same as for linear systems: rearrange the simplest equation and substitute into the other.
Comparison between a linear equation and a quadratic non-linear equation in a system.
x − y = 2
(linear, easy to rearrange)
x² + y² = 34
(non-linear)
When you substitute an expression with two terms inside a square, such as $(y + 2)^2$, do not just add the squares. Always apply the full identity: $(a + b)^2 = a^2 + 2ab + b^2$.
When solving $\begin{cases} x - y = 1 \\ x^2 + y = 7 \end{cases}$, Marcos rearranges $x = y + 1$ and writes in the second equation: $(y + 1)^2 + y = 7 \Rightarrow y^2 + 1 + y = 7$.
Show answer Marcos has forgotten the middle term when squaring the binomial: $(y + 1)^2 = y^2 + 2y + 1$, not $y^2 + 1$.
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Solve the following system of non-linear equations: $ \begin{cases} x - y = 2 \\ x^2 + y^2 = 34 \end{cases} $ Solve the following system of non-linear equations:
$ \begin{cases} x - y = 2 \\ x^2 + y^2 = 34 \end{cases} $
Solve
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Determine the solutions to the system: $ \begin{cases} x^2 - y^2 = 5 \\ x^2 + y^2 = 13 \end{cases} $ Determine the solutions to the system:
$ \begin{cases} x^2 - y^2 = 5 \\ x^2 + y^2 = 13 \end{cases} $
Solve
Line and parabola: points of intersection
Just as 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 whether the line cuts, touches or misses the parabola.
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When you rearrange and substitute you get a quadratic equation: if its discriminant is negative, there is no real solution, even if you have done everything correctly algebraically. Graphically it means that the line does not touch the curve.
Each value of $x$ is paired with its own value of $y$. Substitute each solution separately into the rearranged expression and write the final result as correctly matched ordered pairs $(x, y)$.
When you substitute the line into the parabola you get the equation $x^2 - 2x + 5 = 0$, whose discriminant is $\Delta = (-2)^2 - 4 \cdot 1 \cdot 5 = -16$.
Show answer They have no points of intersection. Since the discriminant is negative, the equation has no real solutions and the line does not meet the parabola.
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Solve the following system with radicals: $ \begin{cases} \sqrt{x} + y = 5 \\ x - y = 1 \end{cases} $ Solve the following system with radicals:
$ \begin{cases} \sqrt{x} + y = 5 \\ x - y = 1 \end{cases} $
Solve
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Solve the system of radical equations: $ \begin{cases} \sqrt{x+y} = 4 \\ x - \sqrt{y} = 2 \end{cases} $ Solve the system of radical equations:
$ \begin{cases} \sqrt{x+y} = 4 \\ x - \sqrt{y} = 2 \end{cases} $
Solve
Sum and product of two numbers
A very common type: you know the sum and the product (or the sum and the difference of squares) of two numbers and you have to find them.
Graph showing the points of intersection between a line and a parabola on the Cartesian plane, illustrating the solutions of a non-linear system.
x + y = 15
&
x · y = 56
Lucía solves a problem in which two numbers add up to 11 and their product is 28. She obtains the solutions $(x = 4, y = 7)$ and $(x = 7, y = 4)$, and states that there are two different pairs of numbers that meet the conditions.
Show answer She is wrong: since the system is symmetric, both numerical solutions correspond to the same pair of numbers, 4 and 7; the only thing that changes is which unknown each value is assigned to.
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Solve the system: $ \begin{cases} x + y = 7 \\ x \cdot y = 12 \end{cases} $ Solve the system:
$ \begin{cases} x + y = 7 \\ x \cdot y = 12 \end{cases} $
Solve
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Calculate two positive integers knowing that their sum is $15$ and their product is $56$. Calculate two positive integers knowing that their sum is $15$ and their product is $56$.
Solve
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The difference between two natural numbers is $3$ and the difference between their squares is $33$. What are these numbers? The difference between two natural numbers is $3$ and the difference between their squares is $33$. What are these numbers?
Solve
Geometry problems: perimeter and area
In rectangles, rhombuses and other shapes, the perimeter usually gives a linear equation and the area a non-linear equation (with a product).
Rectangle with base x and height y and its perimeter and area formulae.
x
y
A = x · y = 48
P = 2x + 2y = 28
When solving a problem about the dimensions of a plot of land you obtain the solutions $(x = 10, y = 4)$ and $(x = 4, y = 10)$.
Show answer No, it is the same rectangle with sides of 10 m and 4 m. The two solutions just swap which side you call the base and which the height.
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The perimeter of a rectangle is $28 \text{ cm}$ and its area is $48 \text{ cm}^2$. Determine the lengths of its sides by setting up a system of equations. The perimeter of a rectangle is $28 \text{ cm}$ and its area is $48 \text{ cm}^2$. Determine the lengths of its sides by setting up a system of equations.
x
y
A = 48 cm²
P = 28 cm
Solve
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The diagonals of a rhombus differ by $4 \text{ cm}$ and its area is $96 \text{ cm}^2$. Calculate the length of both diagonals. The diagonals of a rhombus differ by $4 \text{ cm}$ and its area is $96 \text{ cm}^2$. Calculate the length of both diagonals.
D
d
A = 96
Solve
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Find the dimensions of a right triangle whose hypotenuse measures $13 \text{ cm}$, knowing that if both legs increase their length by $2 \text{ cm}$, the area of the… Find the dimensions of a right triangle whose hypotenuse measures $13 \text{ cm}$, knowing that if both legs increase their length by $2 \text{ cm}$, the area of the triangle increases by $19 \text{ cm}^2$.
Solve
Checking the solutions in context
A non-linear system may give several mathematical solutions, but not all of them are useful in a real problem.
Diagram showing the selection of a valid positive solution versus the rejection of negative solutions in real-life problems.
solutions
x = 6
valid
x = −4
rejected
x = length
→ x = −4 is rejected
If $x$ is a length, a number of people or a price, negative solutions (or non-integer ones, depending on the case) are not valid even though they solve the equations correctly. Always check the question before giving the final answer.
We call the legs $x$ and $y$. We set up the system: $\begin{cases} x - y = 2 \\ x^2 + y^2 = 10^2 \end{cases}$. We rearrange $x = y + 2$ and substitute: $(y + 2)^2 + y^2 = 100$.
We expand and simplify: $y^2 + 4y + 4 + y^2 = 100 \Rightarrow 2y^2 + 4y - 96 = 0 \Rightarrow y^2 + 2y - 48 = 0$. We obtain $y = 6$ or $y = -8$.
If $y = 6$, then $x = 8$. We discard the solution $y = -8$ because a length cannot be negative. The legs measure 6 cm and 8 cm .
Next step (1/3)
When calculating the current ages of two siblings using a system you obtain $(x = 14, y = 9)$ and $(x = -9, y = -14)$.
Show answer Only $(14, 9)$ is valid. The solution with negative values must be discarded because a real age cannot be negative.
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If the base of a rectangle increases by $2 \text{ cm}$ and its height increases by $1 \text{ cm}$, the area increases by $22 \text{ cm}^2$. Knowing that the original… If the base of a rectangle increases by $2 \text{ cm}$ and its height increases by $1 \text{ cm}$, the area increases by $22 \text{ cm}^2$. Knowing that the original base is twice the height, find the initial dimensions of the rectangle.
Solve
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A wire of length $40 \text{ cm}$ is cut into two pieces. Each piece is formed into a square. If the sum of the areas of both squares is $58 \text{ cm}^2$, what is the… A wire of length $40 \text{ cm}$ is cut into two pieces. Each piece is formed into a square. If the sum of the areas of both squares is $58 \text{ cm}^2$, what is the length of each piece of wire?
40 cm
x
y
A₁ + A₂ = 58 cm²
Solve
Summary
A system is non-linear if any equation contains $x^2$, $x \cdot y$, roots, etc. It is solved in the same way: rearrange and substitute .
Substituting usually produces a quadratic equation : it can give two, one or no real solutions.
A line and a parabola meet at 0, 1 or 2 points, depending on the discriminant of the resulting equation.
The classic "sum and product" case of two numbers is always solved this way.
In geometry: the perimeter usually gives the linear equation and the area the non-linear one.
Always check the solutions in the context of the problem: discard those that do not make sense.
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