Question

How many solutions does the system of equations below have?

3x − 8y = 4
6x − 16y = 17

Answer 1

To determine the number of solutions for the given system of equations, we can analyze the two equations:

1. \(3x - 8y = 4\) (Equation 1)
2. \(6x - 16y = 17\) (Equation 2)

First, notice that Equation 2 can be simplified. We can rewrite it by dividing every term by 2:

\[
3x - 8y = \frac{17}{2}
\]

Now we have:

1. \(3x - 8y = 4\) (Equation 1)
2. \(3x - 8y = \frac{17}{2}\) (Simplified Equation 2)

We can see that both equations represent lines in the \(xy\)-plane. Since both have the same left-hand side, they must be equal to each other:

\[
4 = \frac{17}{2}
\]

This equation is not true because \(4\) (which is \(\frac{8}{2}\)) is not equal to \(\frac{17}{2}\).

Since the left-hand sides of both equations are identical but their right-hand sides are different, the two lines represented by these equations are parallel and will never intersect.

Thus, the system of equations has **no solutions**.