To find the area of the triangle formed by the lines y = x, y = 2x, and y = 3x + 4, we can use the concept of determinants.

First, let's find the points of intersection of these lines.

1. Intersection of y = x and y = 2x:
To find the point of intersection between y = x and y = 2x, we can equate the two equations:
x = 2x
This gives us x = 0. Substituting this value back into either equation, we find y = 0 as well. Therefore, the point of intersection is (0, 0).

2. Intersection of y = x and y = 3x + 4:
To find the point of intersection between y = x and y = 3x + 4, we can equate the two equations:
x = 3x + 4
This gives us x = -1. Substituting this value back into either equation, we find y = -1 as well. Therefore, the point of intersection is (-1, -1).

3. Intersection of y = 2x and y = 3x + 4:
To find the point of intersection between y = 2x and y = 3x + 4, we can equate the two equations:
2x = 3x + 4
This gives us x = -4. Substituting this value back into either equation, we find y = -8 as well. Therefore, the point of intersection is (-4, -8).

Now that we have the three points of intersection, we can form a matrix using these points.

Matrix A:
| x1 y1 1 |
| x2 y2 1 |
| x3 y3 1 |

Substituting the values of the points of intersection into the matrix, we get:

Matrix A:
| 0 0 1 |
| -1 -1 1 |
| -4 -8 1 |

To find the area of the triangle, we need to evaluate the determinant of matrix A and take its absolute value.

Area = |det(A)|

Calculating the determinant of matrix A, we get:

det(A) = ((0 * (-1) * 1) + (0 * 1 * 1) + (1 * (-1) * (-8))) - ((1 * (-1) * (-8)) + (0 * (-4) * 1) + (0 * (-1) * 1))
= (0 + 0 + 8) - (-8 + 0 + 0)
= 8 + 8
= 16

Taking the absolute value of the determinant, we have:

Area = |16| = 16

Therefore, the area of the triangle formed by the lines y = x, y = 2x, and y = 3x + 4 is 16.

Hence, the correct answer is option C) 16.