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To show that the equation of a line with **x**- and **y**-intercepts \( a \) and \( b \) can be written in the form:

\[
\frac{x}{a} + \frac{y}{b} = 1,
\]

we can start by recalling the meaning of the intercepts.

### Step-by-Step Proof:

1. **Intercept Form Definition**:
   - The **x-intercept** is the point where the line crosses the x-axis. At this point, \( y = 0 \), and the x-coordinate is \( a \). Thus, the point is \( (a, 0) \).
   - The **y-intercept** is the point where the line crosses the y-axis. At this point, \( x = 0 \), and the y-coordinate is \( b \). Thus, the point is \( (0, b) \).

2. **General Form of a Line**:
   - The general form of the equation of a line is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants.
   
3. **Using the Intercepts**:
   - Substituting \( x = a \) and \( y = 0 \) into the general equation \( Ax + By = C \), we get:
     \[
     A \cdot a + B \cdot 0 = C \quad \Rightarrow \quad A \cdot a = C \quad \Rightarrow \quad C = A \cdot a.
     \]
   - Substituting \( x = 0 \) and \( y = b \) into the equation, we get:
     \[
     A \cdot 0 + B \cdot b = C \quad \Rightarrow \quad B \cdot b = C \quad \Rightarrow \quad C = B \cdot b.
     \]
   
4. **Relating \( A \) and \( B \)**:
   - From these two results, we have \( C = A \cdot a \) and \( C = B \cdot b \). Therefore, \( A \cdot a = B \cdot b = C \).
   - To simplify, we can express the equation as:
     \[
     \frac{A}{C} \cdot x + \frac{B}{C} \cdot y = 1.
     \]
   
5. **Deriving the Required Form**:
   - Dividing through by \( C \), we obtain:
     \[
     \frac{x}{a} + \frac{y}{b} = 1.
     \]
   This is the equation of the line in terms of its x- and y-intercepts, \( a \) and \( b \).

Thus, the line with intercepts \( a \) and \( b \) can be written as \( \frac{x}{a} + \frac{y}{b} = 1 \), as required.

---

Would you like more details or have any further questions?

Here are five related questions to deepen your understanding:

1. How does the slope-intercept form of a line compare to the intercept form?
2. What happens to the equation if either intercept, \( a \) or \( b \), is zero?
3. Can you derive the equation of a line with specific intercepts, say \( a = 4 \) and \( b = 3 \)?
4. How can this form of the line equation be converted into the standard form, \( Ax + By = C \)?
5. What is the geometric significance of the intercepts \( a \) and \( b \) on the coordinate plane?

**Tip:** The intercept form of a line equation is useful for quickly identifying the points where the line crosses the axes, simplifying graphing!

Show that if the x- and y-intercepts of a line are nonzero numbers a and b, then the equation of the line can be written in the form x/a + y/b = 1.

Learn how to derive the equation of a line using the x- and y-intercepts, and show that it can be written in the form x/a + y/b = 1. This problem covers key concepts in linear equations and is suitable for students in Grades 9-12.

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