To write the equation of a line in standard form that has an [tex]\( x \)[/tex]-intercept of [tex]\((-P, 0)\)[/tex] and a [tex]\( y \)[/tex]-intercept of [tex]\((0, R)\)[/tex], we can follow these steps:
1. Identify the intercepts:
- [tex]\( x \)[/tex]-intercept: [tex]\((-P, 0)\)[/tex]
- [tex]\( y \)[/tex]-intercept: [tex]\((0, R)\)[/tex]
2. Determine the slope:
The slope [tex]\( m \)[/tex] of the line is calculated using the formula:
[tex]\[
m = \frac{\text{change in } y}{\text{change in } x} = \frac{R - 0}{0 - (-P)} = \frac{R}{P}
\][/tex]
3. Write the equation in slope-intercept form:
The slope-intercept form of a line is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. Given the slope [tex]\( \frac{R}{P} \)[/tex] and the y-intercept [tex]\( R \)[/tex]:
[tex]\[
y = \frac{R}{P} x + R
\][/tex]
4. Convert to standard form (Ax + By = C):
To convert [tex]\( y = \frac{R}{P} x + R \)[/tex] to standard form, first eliminate the fraction by multiplying through by [tex]\( P \)[/tex]:
[tex]\[
Py = Rx + PR
\][/tex]
Rearrange to get all terms on one side:
[tex]\[
Rx - Py = -PR
\][/tex]
Therefore, the standard form equation with the intercepts provided is:
[tex]\[
Rx - Py = -PR
\][/tex]
5. Compare with the given choices:
- [tex]\( R x - P y = -P R \)[/tex]
- [tex]\( R x + P y = P R \)[/tex]
- [tex]\( P x - R y = -P R \)[/tex]
- [tex]\( P x - R y = P R \)[/tex]
The equation [tex]\( Rx - Py = -PR \)[/tex] matches the first choice: [tex]\( R x - P y = -P R \)[/tex].
Hence, the correct answer is:
[tex]\[
\boxed{R x - P y = -P R}
\][/tex]
