# How many solutions does the system {x=10y { y=sin x have?

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The solution of the equations x = 10y and y = sin(x) have to be determined.

x = 10y

y = x/10

Substitute y = x/10 in y = sin(x).

x/10 = sin x

x = 10*sin x

This looks like a simple equation but there is no algebraic way to determine the roots.

One way to find the roots is by plotting y = x and y = 10*sin x and looking at the points of intersection.

From the graph it can be seen that x = 0 is a solution of the equation. We know that sin 0 = 0.

The equation x = sin x has only one root.

But x = 10*sin x has a larger number of solutions as can be seen in the graph. There are 7 points at which the two graphs intersect each other.

The system of equations x=10y and y=sin(x) has five solutions. We plot graphs of these two equations and find that these graphs are intersecting each other at five points.