if a line is neither vertical nor horizontal and does not pass through the origin, show that its equation can be written in the form of (x/a)+(y/b)=1 where a is its x intercept and b is its y intercept

To prove that the equation of a line that is neither vertical nor horizontal and does not pass through the origin can be written in the form \(\frac{x}{a} + \frac{y}{b} = 1\), where \(a\) is its x-intercept and \(b\) is its y-intercept, we can follow these steps:

### Step 1: General Equation of a Line

The general equation of a line in slope-intercept form is:

\[
y = mx + c
\]

where \(m\) is the slope of the line, and \(c\) is the y-intercept.

### Step 2: Find the X-Intercept and Y-Intercept

1. **X-Intercept (\(a\))**: The x-intercept is the point where the line crosses the x-axis. At the x-axis, \(y = 0\). So, set \(y = 0\) in the equation:

   \[
   0 = mx + c \implies x = -\frac{c}{m}
   \]

   Thus, the x-intercept \(a = -\frac{c}{m}\).

2. **Y-Intercept (\(b\))**: The y-intercept is already given by \(c\), so \(b = c\).

### Step 3: Express the Line in Terms of Intercepts

We can use the intercept form of a line, which states that if a line has intercepts \(a\) and \(b\), its equation can be written as:

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

Substituting the intercepts found earlier:

\[
\frac{x}{-\frac{c}{m}} + \frac{y}{c} = 1
\]

Simplify this:

\[
-\frac{mx}{c} + \frac{y}{c} = 1
\]

Multiplying through by \(c\):

\[
-mx + y = c
\]

Rearrange it back to the general form:

\[
y = mx + c
\]

### Conclusion

Since we started with \(y = mx + c\) and expressed it as \(\frac{x}{a} + \frac{y}{b} = 1\) with \(a = -\frac{c}{m}\) and \(b = c\), we have shown that any line which is neither vertical nor horizontal and does not pass through the origin can indeed be written in this intercept form.

Would you like to go over any specific details or have any questions? Here are some related questions you might find interesting:

1. How can you find the slope of a line given two points?
2. What happens if the line passes through the origin in terms of the intercept form?
3. Can a vertical line be expressed in intercept form? Why or why not?
4. How do the intercepts of a line change if it is translated up or down?
5. How does the slope-intercept form compare with the intercept form of a line?

**Tip:** The intercept form is especially useful for quickly finding where the line crosses the axes, making it easy to sketch graphs of lines!

Discover how to prove that a line, neither vertical nor horizontal and not passing through the origin, can be expressed in intercept form. This guide explores key algebraic concepts like slope-intercept form and x and y intercepts, suitable for Grades 9-11.

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