Find the area bounded by the lines `y=4x+5, y=5-x" and "4y=x+5`.

Explanation

To find the area bounded by the lines \(y = 4x + 5\), \(y = 5 - x\), and \(4y = x + 5\), we will follow these steps:

### Step 1: Find the intersection points of the lines
We need to find the intersection points of the three lines to determine the vertices of the bounded area.

1. **Intersection of \(y = 4x + 5\) and \(y = 5 - x\)**:
\[ 4x + 5 = 5 - x \]
\[ 4x + x = 5 - 5 \]
\[ 5x = 0 \implies x = 0 \]
Substituting \(x = 0\) into \(y = 5 - x\):
\[ y = 5 - 0 = 5 \]
So, the intersection point is \((0, 5)\).

2. **Intersection of \(y = 4x + 5\) and \(4y = x + 5\)**:
First, rewrite \(4y = x + 5\) as \(y = \frac{x + 5}{4}\).
\[ 4x + 5 = \frac{x + 5}{4} \]
Multiply through by 4 to eliminate the fraction:
\[ 16x + 20 = x + 5 \]
\[ 16x - x = 5 - 20 \]
\[ 15x = -15 \implies x = -1 \]
Substituting \(x = -1\) into \(y = 4x + 5\):
\[ y = 4(-1) + 5 = -4 + 5 = 1 \]
So, the intersection point is \((-1, 1)\).

3. **Intersection of \(y = 5 - x\) and \(4y = x + 5\)**:
Using \(y = \frac{x + 5}{4}\):
\[ 5 - x = \frac{x + 5}{4} \]
Multiply through by 4:
\[ 20 - 4x = x + 5 \]
\[ 20 - 5 = x + 4x \]
\[ 15 = 5x \implies x = 3 \]
Substituting \(x = 3\) into \(y = 5 - x\):
\[ y = 5 - 3 = 2 \]
So, the intersection point is \((3, 2)\).

### Step 2: Identify the vertices of the bounded region
The vertices of the bounded area are \((0, 5)\), \((-1, 1)\), and \((3, 2)\).

### Step 3: Calculate the area using the formula
The area \(A\) of a polygon given vertices \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\) can be calculated using the formula:
\[ A = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_1 - (y_1x_2 + y_2x_3 + y_3x_1) \right| \]

Substituting the vertices:
\[ A = \frac{1}{2} \left| 0 \cdot 1 + (-1) \cdot 2 + 3 \cdot 5 - (5 \cdot (-1) + 1 \cdot 3 + 2 \cdot 0) \right| \]
Calculating each term:
\[ = \frac{1}{2} \left| 0 - 2 + 15 - (-5 + 3 + 0) \right| \]
\[ = \frac{1}{2} \left| 0 - 2 + 15 + 5 - 3 \right| \]
\[ = \frac{1}{2} \left| 15 - 2 + 5 - 3 \right| \]
\[ = \frac{1}{2} \left| 15 - 2 + 5 - 3 \right| = \frac{1}{2} \left| 15 \right| = \frac{15}{2} \]

### Final Answer
Thus, the area bounded by the lines is \(\frac{15}{2}\) square units.

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