Simultaneous Equations Graphs

Maths: Edexcel A Level Pure Maths, topic 2.5.2

Solving Nonlinear Systems of Equations

A system of nonlinear equations is a system of two or more equations in two or more variables containing at least one equation that is not linear.

There are three possible types of solutions for a system of nonlinear equations involving a parabola and a line.
The points where the lines cross are solutions to both equations.

In the image, the functions $y = x^2 + 1$ and $x-y=-1$ are plotted.
The points of intersection are $(0,1)$ and $(1,2)$ , so the solutions of this system of equations are:
$x=0, y=1$ and $x=1,y=2$

Just as with a parabola and a line, there are three possible outcomes when solving a system of equations representing a circle and a line.
The line that has one solution is tangent to the circle and intersects the circle at exactly one point.

In the image, the functions $x^2+y^2 = 5$ and $y=3x-5$ are plotted.
The points of intersection are $(2,1)$ and $(1,-2)$ , so the solutions of this system of equations are:
$x=2, y=1$ and $x=1,y=-2$

Using the graph, what are the solutions to the following sytem of equations:
y = x, y = x² - 2
How many solutions are there to the system of equations shown on the graph:
y = x² - 2 (blue) and y = x (red)
How many solutions are there to the system of equations shown on the graph:
y = x² - 4 (red) and y = x + 2 (blue)
Using the graph, what are the solutions to the following system of equations:
y = x² - 4, y = x - 2
What is the solution to the system of equations shown by the graph?