To find the equation of a line in standard form that has specific intercepts, we must follow several steps. The intercepts provided are [tex]\( x \)[/tex]-intercept [tex]\((-P, 0)\)[/tex] and [tex]\( y \)[/tex]-intercept [tex]\((0, R)\)[/tex].
1. Interpret the intercepts:
- The [tex]\( x \)[/tex]-intercept [tex]\((-P, 0)\)[/tex] means the line crosses the [tex]\( x \)[/tex]-axis at [tex]\((-P, 0)\)[/tex].
- The [tex]\( y \)[/tex]-intercept [tex]\((0, R)\)[/tex] means the line crosses the [tex]\( y \)[/tex]-axis at [tex]\((0, R)\)[/tex].
2. Equation in intercept form:
The equation of a line in intercept form is given by:
[tex]\[
\frac{x}{-P} + \frac{y}{R} = 1
\][/tex]
3. Convert the intercept form to standard form [tex]\( Ax + By = C \)[/tex]:
- To eliminate the denominators, we multiply through by [tex]\(-P \cdot R\)[/tex]:
[tex]\[
R \cdot x + (-P) \cdot y = -P \cdot R
\][/tex]
4. Simplify the equation:
- Arrange the equation in the standard form:
[tex]\[
R \cdot x - P \cdot y = -P \cdot R
\][/tex]
5. Identify coefficients [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex]:
- Here:
[tex]\( A = R \)[/tex]
[tex]\( B = -P \)[/tex]
[tex]\( C = -P \cdot R \)[/tex]
Therefore, the equation of the line in standard form is:
[tex]\[
R x - P y = -P R
\][/tex]
Given the provided numerical solution leads us to identify that:
[tex]\[
1 x - 1 y = -1
\][/tex]
So, the particular multiple-choice option that matches this form is:
[tex]\[
R x - P y = -P R
\][/tex]
