For each linear equation, given above, draw the graph and then use the graph drawn (in each case)to find the area of a triangle enclosed by the graph and the co-ordinate axes:
(i) `3x-(5-y)=7`

Answer

To solve the problem step by step, we will first rewrite the given linear equation, find the intercepts, plot the graph, and then calculate the area of the triangle formed by the graph and the coordinate axes.

### Step 1: Rewrite the equation
The given equation is:
\[ 3x - (5 - y) = 7 \]

First, we simplify this equation:
\[ 3x - 5 + y = 7 \]
Rearranging gives us:
\[ y + 3x = 12 \]
Thus, the equation can be rewritten as:
\[ y = -3x + 12 \]

### Step 2: Find the x-intercept and y-intercept
To find the intercepts, we will set \(x\) and \(y\) to zero in turn.

**Finding the y-intercept:**
Set \(x = 0\):
\[ y = -3(0) + 12 = 12 \]
So, the y-intercept is at the point \((0, 12)\).

**Finding the x-intercept:**
Set \(y = 0\):
\[ 0 = -3x + 12 \]
Solving for \(x\):
\[ 3x = 12 \implies x = 4 \]
So, the x-intercept is at the point \((4, 0)\).

### Step 3: Plot the graph
Now, we can plot the points \((0, 12)\) and \((4, 0)\) on a coordinate plane. Draw a straight line through these points to represent the equation \(y = -3x + 12\).

### Step 4: Identify the triangle formed
The triangle is formed by the line we just drew and the coordinate axes. The vertices of the triangle are:
- \((0, 0)\) (the origin)
- \((0, 12)\) (y-intercept)
- \((4, 0)\) (x-intercept)

### Step 5: Calculate the area of the triangle
The area \(A\) of a right-angled triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
In our triangle:
- The base is the distance along the x-axis from \((0, 0)\) to \((4, 0)\), which is \(4\) units.
- The height is the distance along the y-axis from \((0, 0)\) to \((0, 12)\), which is \(12\) units.

Substituting these values into the area formula:
\[ A = \frac{1}{2} \times 4 \times 12 = \frac{48}{2} = 24 \text{ square units} \]

### Final Answer
The area of the triangle enclosed by the graph and the coordinate axes is \(24\) square units.

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