How many solutions does the linear system y = 5x + 1 and 15x + 3y = 3 have?

To determine how many solutions the given linear system has, we need to analyze the two equations:

Now, let’s substitute the first equation into the second equation to see if we can find any values of x and y:

Substituting for y in the second equation gives:

15x + 3(5x + 1) = 3

Simplifying this:

15x + 15x + 3 = 3

30x + 3 = 3

30x = 3 - 3

30x = 0

x = 0

Now that we have the value of x, we can substitute it back into the first equation to find y:

y = 5(0) + 1

y = 1

So, we have found a single solution to the system of equations, which is (x, y) = (0, 1).

We can also analyze the nature of these equations further:

The first equation is a straightforward linear equation representing a line on the Cartesian plane.
The second equation can be rewritten in slope-intercept form as y = -5x + 1 (by rearranging it), which also represents a line.

Since both equations represent lines and we have found a single point where they intersect, we can conclude that:

This linear system has exactly one solution.