The standard equation for a line is :
y = ax + b
y is called the dependent variable because its value depends on the value of x
a is the slope of the line
x is called the independent variable because it can be any value
b is the y-intercept
In your case : y(x) = 5x, y(x) is just another way of writing the equation to indicate that y is a function of x. You could simply write it as y = 5x. Thus if we compare to the standard form equation y = ax + b, a=5 and b=0. Thus the y-inercept is zero. The reason that this is the case is that when you set x=0 the value is plotted on the y axis, because the equation for the y axis is the line defined by x=0. Any value in the x,y plane where x is equal to zero plots on the vertical line going through (0,0) or in other words the y axis.
The x-intercept is a little harder to find. We have to set y=0 as that is the line defining the x axis. We start with the original equation. y = ax + b. Setting y=0 gives ax+b = 0. Subtracting b from both sides gives ax = -b. Finally dividing both sides by a yields the x-intercept, x = -b/a.
From above for y=5x, a=5 and b=0, the the x-intercept would be -0/5 which is also 0. So when b=0 the line goes through (0,0) otherwise known as the origin.