Number and nature of solutions to non-linear systems

So far, we have looked at solving systems of linear equations, such as the pair of equations $y=x+1$y=x+1 and $y=4-x$y=4−x. These are not the only kinds of systems of equations that we might come across, however.

 

Exploration

The Earth's orbit around the Sun is close to a circle in shape, and we can model this with the equation $x^2+y^2=150^2$x2+y2=1502 (where the units are in millions of km). A stray comet is passing near to the sun, and follows a path given by the equation $y=\frac{x^2}{240}-60$y=x2240​−60. How many times does the comet's path cross the Earth's path around the sun?

Recall that the real solutions to a system of linear equations can be thought of as the points of intersection of their graphs. For a system of two linear equations, there are three possibilities:

• If the two lines are not parallel, such as $y=x+1$y=x+1 and $y=4-x$y=4−x, they intersect at one point. This type of system has one real solution.
• If the two lines are parallel and distinct, such as $y=x+1$y=x+1 and $y=x$y=x, they never intersect. This type of system has no real solutions.
• If the two lines are equivalent, such as $y=x+1$y=x+1 and $2y=2x+2$2y=2x+2, they overlap and intersect at every point. This type of system has infinitely many real solutions.

We can take the same approach for any system of equations. By drawing the curves on the same coordinate plane, we can identify the number of real solutions by looking at the number of points of intersection.

Here is a graph of the equations $x^2+y^2=150^2$x2+y2=1502 and $y=\frac{x^2}{240}-60$y=x2240​−60:

We can immediately see from this graph that there are two real solutions to the system $x^2+y^2=150^2$x2+y2=1502 and $eq=2$eq=2. That is, there are two places where the comet's path crosses the Earth's orbit.

Of course, as long as the Earth is not at those specific points at the same time as the comet, there won't be any collisions.

 

For systems of non-linear equations there are different sets of possible solutions, depending on the types of equations involved. Importantly, systems of non-linear equations can still have:

• no real solutions, when the graphs have no points of intersection.
• a finite number of real solutions (one or more), when the graphs intersect a finite number of times.
• infinitely many real solutions, which most commonly happens when the two equations are identical after rearrangement and intersect at every point.

Here are examples of each case.

No real solutions

The system of equations $y=x^2+1$y=x2+1 and $y=-x^2-1$y=−x2−1 has no real solutions. We can see that the graphs have no points of intersection:

Finitely many real solutions

The system of equations $y=x^2$y=x2 and $x^2+y^2=4$x2+y2=4 has two real solutions. We can see that their graphs have two points of intersection:

Infinitely many real solutions

The system of equations $y=\frac{2}{x-1}$y=2x−1​ and $x=\frac{2}{y}+1$x=2y​+1 has infinitely many real solutions. We could rearrange one equation to obtain the other, and we can see that their graphs are identical (and so they intersect at every point):

 

 

Practice questions

Question 1

Question 2

Question 3