Nonlinear systems of equations are not just for hypothetical discussions—they can be used to solve complex problems involving multiple known relationships.

Real World Examples

Consider, for example, a car that begins at rest and accelerates at a constant rate of $4$ meters per second each second. Its position in meters ($y$) can be determined as a function of time in seconds ($t$), by the formula:

$\displaystyle{ \begin{aligned} y&=\frac{1}{2}\left(4\right)t^2\\ y&=2t^2 \end{aligned} }$ 

Now consider a second car, traveling at a constant speed of $20$ meters per second. Its position ($y$) in meters can be determined as a function of time ($t$) in seconds, using the following formula:

$y=20t$ 

When the first car begins to accelerate, the second car is $400$ meters ahead of it. To express the position of the second car relative to the first as a function of time, we can modify the second equation as such:

$y=20t+400$ 

To determine where the cars are when they are alongside one another and how much time has passed since the first began to accelerate, we can algebraically solve the system of equations using substitution:

$\begin{aligned} y&=20t+400\\ 2t^2&=20t+400 \end{aligned}$ 

Solving for $t$, we can find that the cars are side-by-side after $20$ seconds.

Substituting $20$ for $t$ into the equations for either of the cars, we can find that the cars meet $800$ meters ahead of the first car's starting point. Note that a question on an exam may not prompt solutions for both variables.

 Some other real-world examples of nonlinear systems include:

• Triangulation of GPS signals. A device like your cellphone receives signals from GPS satellites, which have known orbital positions around the Earth. A signal from a single satellite allows a cellphone to know that it is somewhere on a circle. Additional signals are additional circles that intersect each other, and the cellphone's actual position is at the intersection. Three or more signals reduce the solution of the system to a single coordinate point.
• The conservation of mechanical energy can produce a system of nonlinear equations when there is an elastic (perfectly bouncy) collision. The kinetic energy of the objects depends on the speed squared, and the momentum depends on the speed directly.
• Manufacturing and design of everything, from electronic parts to metal tools to the architecture of buildings, uses computer-aided design software that helps create three-dimensional shapes from the intersection of curved lines. Rendering and visualizing these objects, and formulating a plan for constructing them, requires the software to solve nonlinear systems. 

Additional Example

In addition to practical scenarios like the above, nonlinear systems can be used in abstract problems. For example, a question on an exam could ask:

The product of two numbers is 12, and the sum of their squares is 40. What are the numbers?

In this case, we could make an equation for each known relationship:

$\begin{aligned} x\cdot y&=12 \\ x^2+y^2&=40 \end{aligned}$ 

Substitution can be used to calculate that the numbers are 2 and 6.