Finding the Area of the Triangle Formed by Linear Equations and the X-axis

The problem asks us to find the area of a triangle formed by the graphs of three lines: $8x + 3y = 24$, $2x + 8 = y$, and the x-axis. To find the area of this triangle, we first need to determine the coordinates of its vertices. The vertices are the points where these lines intersect each other.

Identifying the Lines

We are given three lines:

1. Line 1: $8x + 3y = 24$
2. Line 2: $y = 2x + 8$
3. Line 3: The x-axis, which is represented by the equation $y = 0$.

Finding the Vertices of the Triangle

The vertices are the intersection points of these lines:

Vertex 1: Intersection of Line 1 ($8x + 3y = 24$) and the x-axis ($y = 0$)

Substitute $y = 0$ into the equation $8x + 3y = 24$:

$8x + 3(0) = 24$
$8x = 24$
$x = \frac{24}{8}$
$x = 3$

The first vertex is $(3, 0)$.

Vertex 2: Intersection of Line 2 ($y = 2x + 8$) and the x-axis ($y = 0$)

Substitute $y = 0$ into the equation $y = 2x + 8$:

$0 = 2x + 8$
$-8 = 2x$
$x = \frac{-8}{2}$
$x = -4$

The second vertex is $(-4, 0)$.

Vertex 3: Intersection of Line 1 ($8x + 3y = 24$) and Line 2 ($y = 2x + 8$)

Substitute the expression for $y$ from Line 2 into the equation for Line 1:

$8x + 3(2x + 8) = 24$
$8x + 6x + 24 = 24$
$14x + 24 = 24$
$14x = 24 - 24$
$14x = 0$
$x = 0$

Now, substitute the value of $x$ back into the equation for Line 2 to find $y$:

$y = 2(0) + 8$
$y = 0 + 8$
$y = 8$

The third vertex is $(0, 8)$.

Calculating the Area of the Triangle

The vertices of the triangle are $(3, 0)$, $(-4, 0)$, and $(0, 8)$.

Notice that two vertices, $(3, 0)$ and $(-4, 0)$, lie on the x-axis ($y=0$). This forms the base of our triangle along the x-axis.

The length of the base is the distance between $(3, 0)$ and $(-4, 0)$.

Base length $= |3 - (-4)| = |3 + 4| = 7$ units.

The height of the triangle is the perpendicular distance from the third vertex $(0, 8)$ to the base (the x-axis). This distance is simply the absolute value of the y-coordinate of the third vertex.

Height $= |8| = 8$ units.

The area of a triangle is given by the formula:

Area $= \frac{1}{2} \times \text{base} \times \text{height}$

Substitute the base and height values:

Area $= \frac{1}{2} \times 7 \times 8$

Area $= \frac{1}{2} \times 56$

Area $= 28$ square units.

Thus, the area of the triangle formed by the given lines and the x-axis is 28 square units.

Revision Table: Key Steps to Find Triangle Area from Lines

Additional Information: Graphing Linear Equations

Graphing linear equations like $8x + 3y = 24$ and $y = 2x + 8$ can help visualize the triangle. To graph a linear equation, you can find two points that satisfy the equation and draw a line through them. Common points to find are the x-intercept (where $y=0$) and the y-intercept (where $x=0$).

• For $8x + 3y = 24$:
  • If $x=0$, $3y=24 \implies y=8$. Point: $(0, 8)$.
  • If $y=0$, $8x=24 \implies x=3$. Point: $(3, 0)$.
• For $y = 2x + 8$:
  • If $x=0$, $y=8$. Point: $(0, 8)$.
  • If $y=0$, $0=2x+8 \implies 2x=-8 \implies x=-4$. Point: $(-4, 0)$.

Notice that the point $(0, 8)$ is common to both lines, which is one of the vertices we found. The points $(3, 0)$ and $(-4, 0)$ are the x-intercepts of the two lines, which form the base on the x-axis ($y=0$). Plotting these points and lines confirms the triangle vertices.