• Question Analysis: The problem asks us to find the area of a specific region in the coordinate plane. This region is defined as the space enclosed by the graphs of three linear equations: $2x - 3y + 6 = 0$, $4x + y = 16$, and $y = 0$.
• Keywords: Area, region enclosed, graphs, equations, $2x - 3y + 6 = 0$, $4x + y = 16$, $y = 0$.

Defining the Boundary Equations

To find the area of the enclosed region, we first need to clearly identify the lines that form its boundaries:

• Line 1: $2x - 3y + 6 = 0$
• Line 2: $4x + y = 16$
• Line 3: $y = 0$. This equation represents the x-axis.

Calculating the Vertices of the Region

The region enclosed by these lines forms a polygon. The corners (vertices) of this region are found by determining where these lines intersect each other.

1. Intersection of Line 1 ($2x - 3y + 6 = 0$) and Line 3 ($y = 0$): Substitute $y = 0$ into the first equation: $2x - 3(0) + 6 = 0$ $2x + 6 = 0$ $2x = -6$ $x = -3$ So, the first vertex is A = (-3, 0).
2. Intersection of Line 2 ($4x + y = 16$) and Line 3 ($y = 0$): Substitute $y = 0$ into the second equation: $4x + 0 = 16$ $4x = 16$ $x = 4$ So, the second vertex is B = (4, 0).
3. Intersection of Line 1 ($2x - 3y + 6 = 0$) and Line 2 ($4x + y = 16$): First, express $y$ from Line 2: $y = 16 - 4x$. Now, substitute this expression for $y$ into Line 1: $2x - 3(16 - 4x) + 6 = 0$ $2x - 48 + 12x + 6 = 0$ Combine like terms: $14x - 42 = 0$ $14x = 42$ $x = \frac{42}{14} = 3$ Now find the corresponding $y$ value using $y = 16 - 4x$: $y = 16 - 4(3)$ $y = 16 - 12$ $y = 4$ So, the third vertex is C = (3, 4).

The vertices of the enclosed region are (-3, 0), (4, 0), and (3, 4). These points form a triangle.

Determining the Area of the Triangle

The area of the enclosed region, which is a triangle with vertices A(-3, 0), B(4, 0), and C(3, 4), can be calculated using the base and height.

• Base: The base of the triangle lies along the x-axis (where $y = 0$) between vertex A (-3, 0) and vertex B (4, 0). The length of the base is the distance between these two points: $\text{Base} = |4 - (-3)| = |4 + 3| = 7$ units.
• Height: The height of the triangle is the perpendicular distance from the third vertex C (3, 4) to the base (the x-axis). This distance is simply the absolute value of the y-coordinate of vertex C. $\text{Height} = |4| = 4$ units.
• Area Calculation: The formula for the area of a triangle is $\frac{1}{2} \times \text{base} \times \text{height}$. $\text{Area} = \frac{1}{2} \times 7 \times 4$ $\text{Area} = \frac{1}{2} \times 28$ $\text{Area} = 14$ unit squares.

Conclusion on the Region's Area

The calculated area of the region enclosed by the graphs of the equations $2x - 3y + 6 = 0$, $4x + y = 16$, and $y = 0$ is 14 unit squares.