`(x)/(a)+(y)/(b)=1` is intercept from of line. (Prove It)

Answer

To prove that the equation \(\frac{x}{a} + \frac{y}{b} = 1\) is the intercept form of a line, we can follow these steps:

### Step 1: Understand the Intercept Form of a Line
The intercept form of a line is generally expressed as:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
where \(a\) is the x-intercept (the point where the line intersects the x-axis) and \(b\) is the y-intercept (the point where the line intersects the y-axis).

### Step 2: Identify the Intercepts
From the equation \(\frac{x}{a} + \frac{y}{b} = 1\):
- When \(y = 0\) (to find the x-intercept), we substitute \(y\) into the equation:
\[ \frac{x}{a} + \frac{0}{b} = 1 \implies \frac{x}{a} = 1 \implies x = a \]
Thus, the x-intercept is \((a, 0)\).

- When \(x = 0\) (to find the y-intercept), we substitute \(x\) into the equation:
\[ \frac{0}{a} + \frac{y}{b} = 1 \implies \frac{y}{b} = 1 \implies y = b \]
Thus, the y-intercept is \((0, b)\).

### Step 3: Plot the Points
Now, we can plot the points \((a, 0)\) and \((0, b)\) on the coordinate axis. These points will help us visualize the line.

### Step 4: Use the Two-Point Form of a Line
The two-point form of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ \frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1} \]
In our case, we can denote:
- \((x_1, y_1) = (a, 0)\)
- \((x_2, y_2) = (0, b)\)

Substituting these points into the two-point form:
\[ \frac{y - 0}{b - 0} = \frac{x - a}{0 - a} \]
This simplifies to:
\[ \frac{y}{b} = \frac{x - a}{-a} \]

### Step 5: Rearranging the Equation
Cross-multiplying gives:
\[ y = -\frac{b}{a}(x - a) \]
Expanding this:
\[ y = -\frac{b}{a}x + b \]

### Step 6: Rearranging to the Intercept Form
Now, we can rearrange this equation:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
This shows that the equation \(\frac{x}{a} + \frac{y}{b} = 1\) indeed represents a line in intercept form.

### Conclusion
Thus, we have proven that the equation \(\frac{x}{a} + \frac{y}{b} = 1\) is the intercept form of a line.

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