x+4y=3, 2x+8y=4 The system of equations above has how many solutions?

Hmm, it seems we have a case of "serious equations." Let me grab my clown nose and investigate!

Well, these equations seem like two parallel lines that can never meet. So, it looks like we have a whopping total of... drumroll, please... no solutions! Yep, it seems these equations are set to lead separate lives forever.

To determine the number of solutions of a system of equations, we can use the concept of linear independence or dependence.

Let's rewrite the system of equations in matrix form:

```
| 1 4 | | x | | 3 |
| 2 8 | * | y | = | 4 |
```

Calculating the determinant of the coefficient matrix (the 2x2 matrix on the left), we have:

(1 * 8) - (4 * 2) = 8 - 8 = 0

Since the determinant is equal to 0, the coefficient matrix is linearly dependent.

Now, let's examine the right side of the equation. The right side is not equal to (0,0), which means it is linearly independent.

When the coefficient matrix is linearly dependent (determinant = 0) and the right side is linearly independent, the system of equations has no unique solution. Instead, it can have infinitely many solutions or no solutions at all.

In this case, the system of equations has infinitely many solutions.

To determine the number of solutions in a system of equations, you need to use the concept of intersecting lines. If the lines intersect at a single point, the system has one solution. If the lines are parallel and do not intersect, the system has no solutions. Lastly, if the lines coincide and overlap each other, the system has infinitely many solutions.

In this case, let's rearrange the equations into slope-intercept form (y = mx + b):

Equation 1: x + 4y = 3
Rearranging, we get: 4y = -x + 3
Dividing by 4, we have: y = -1/4x + 3/4

Equation 2: 2x + 8y = 4
Rearranging, we get: 8y = -2x + 4
Dividing by 8, we have: y = -1/4x + 1/2

By comparing both equations, we can see that they have the same slope (-1/4) but different y-intercepts (3/4 and 1/2). This implies that the lines are parallel and will never intersect.

Therefore, since the lines are parallel and do not intersect, the system of equations has no solutions.

since the lines have the same slope, they are either parallel (no solutions) or the same line (infinitely many solutions)

divide the 2nd equation by 2, and it should become clear.