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.
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:
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:
Here are examples of each case.
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:
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:
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):
The real solutions to a system of equations can be thought of as the points of intersection of their graphs. So to determine the number of solutions to a particular system, we can sketch the graphs and see how many times they intersect!
Depending on the particular equations, the system might have no real solutions, a finite number of real solutions (one or more), or infinitely many real solutions.
A graph of the equations $y=x+2$y=x+2 and $y=x^2$y=x2 is shown below.
How many real solutions does this system of equations have?
A graph of the equations $y=x^2-3x+1$y=x2−3x+1 and $y=-x^2-3x+1$y=−x2−3x+1 is shown below.
How many real solutions does this system of equations have?
Determine the number of solutions to the system of equations $y=x\left(x+3\right)$y=x(x+3) and $y=-3x\left(x+3\right)$y=−3x(x+3).
$0$0
$1$1
$2$2
$\infty$∞