Finding the Area of the Enclosed Triangle

The problem asks for the area of the triangle enclosed by the graphs of three lines: 2x + 5y = 12, x + y = 3, and the x-axis. The x-axis is represented by the equation y = 0.

To find the area of the triangle, we first need to identify its vertices. The vertices are the points where these three lines intersect.

Step 1: Find the Vertices of the Triangle

There are three vertices, corresponding to the intersections of the three pairs of lines:

1. Intersection of 2x + 5y = 12 and the x-axis (y = 0).
2. Intersection of x + y = 3 and the x-axis (y = 0).
3. Intersection of 2x + 5y = 12 and x + y = 3.

Finding Vertex 1: Intersection of 2x + 5y = 12 and y = 0

Substitute $y = 0$ into the first equation:

$$2x + 5(0) = 12$$ $$2x = 12$$ $$x = \frac{12}{2} = 6$$

So, Vertex 1 is (6, 0).

Finding Vertex 2: Intersection of x + y = 3 and y = 0

Substitute $y = 0$ into the second equation:

$$x + 0 = 3$$ $$x = 3$$

So, Vertex 2 is (3, 0).

Finding Vertex 3: Intersection of 2x + 5y = 12 and x + y = 3

We have a system of two linear equations:

• Equation 1: $2x + 5y = 12$
• Equation 2: $x + y = 3$

From Equation 2, we can express x in terms of y:

$$x = 3 - y$$

Now, substitute this expression for x into Equation 1:

$$2(3 - y) + 5y = 12$$ $$6 - 2y + 5y = 12$$ $$6 + 3y = 12$$ $$3y = 12 - 6$$ $$3y = 6$$ $$y = \frac{6}{3} = 2$$

Now, substitute the value of y back into the expression for x:

$$x = 3 - y = 3 - 2 = 1$$

So, Vertex 3 is (1, 2).

Step 2: Identify the Base and Height of the Triangle

The three vertices of the triangle are (6, 0), (3, 0), and (1, 2).

Two of the vertices, (6, 0) and (3, 0), lie on the x-axis. This segment on the x-axis forms the base of the triangle. The length of the base is the distance between these two points:

$$Base = |6 - 3| = 3 \text{ units}$$

The height of the triangle is the perpendicular distance from the third vertex, (1, 2), to the base (the x-axis). This distance is the absolute value of the y-coordinate of the third vertex.

$$Height = |2| = 2 \text{ units}$$

Step 3: Calculate the Area of the Triangle

The area of a triangle is given by the formula:

$$Area = \frac{1}{2} \times Base \times Height$$

Using the base and height we found:

$$Area = \frac{1}{2} \times 3 \times 2$$ $$Area = \frac{1}{2} \times 6$$ $$Area = 3 \text{ square units}$$

The area of the triangle enclosed by the given lines and the x-axis is 3 unit squares.

Revision Table: Triangle Area Calculation

Additional Information: Lines and Triangles

Lines in a coordinate plane can intersect to form geometric shapes like triangles. Finding the vertices is crucial for calculating the area or other properties of the shape. When one of the bounding lines is an axis (like the x-axis, y=0), it often simplifies finding the base or height of the triangle.

• A linear equation in two variables (like ax + by = c) represents a straight line in the coordinate plane.
• The intersection point of two lines is the solution to the system of equations representing those lines.
• The distance between two points $(x_1, 0)$ and $(x_2, 0)$ on the x-axis is $|x_2 - x_1|$.
• The distance from a point $(x, y)$ to the x-axis is $|y|$.

Understanding how to solve systems of linear equations and basic coordinate geometry concepts is fundamental to solving this type of problem.