To find the [tex]\(x\)[/tex]-coordinate of point [tex]\(Q\)[/tex], given that point [tex]\(R\)[/tex] divides the line segment [tex]\(\overline{PQ}\)[/tex] in the ratio [tex]\(1:3\)[/tex] and knowing the [tex]\(x\)[/tex]-coordinates of [tex]\(R\)[/tex] and [tex]\(P\)[/tex], we can use the section formula for internal division.
The section formula for internal division states that if a point [tex]\(R\)[/tex] divides the line segment [tex]\(\overline{PQ}\)[/tex] in the ratio [tex]\(m:n\)[/tex], the [tex]\(x\)[/tex]-coordinate of [tex]\(R\)[/tex] can be given by:
[tex]\[
x_R = \frac{m \cdot x_Q + n \cdot x_P}{m + n}
\][/tex]
Here, the given values are:
- [tex]\(x_R = -1\)[/tex]
- [tex]\(x_P = -3\)[/tex]
- The ratio [tex]\(m:n = 1:3\)[/tex], which implies [tex]\(m = 1\)[/tex] and [tex]\(n = 3\)[/tex]
Substituting the known values into the section formula, we have:
[tex]\[
-1 = \frac{1 \cdot x_Q + 3 \cdot (-3)}{1 + 3}
\][/tex]
Simplify the denominator:
[tex]\[
-1 = \frac{x_Q + 3(-3)}{4}
\][/tex]
Next, simplify inside the numerator:
[tex]\[
-1 = \frac{x_Q - 9}{4}
\][/tex]
To eliminate the fraction, multiply both sides by 4:
[tex]\[
4 \cdot (-1) = x_Q - 9
\][/tex]
[tex]\[
-4 = x_Q - 9
\][/tex]
Solve for [tex]\(x_Q\)[/tex]:
[tex]\[
-4 + 9 = x_Q
\][/tex]
[tex]\[
x_Q = 5
\][/tex]
Thus, the [tex]\(x\)[/tex]-coordinate of [tex]\(Q\)[/tex] is [tex]\(5\)[/tex]. Therefore, the correct answer is:
[tex]\[
\boxed{5}
\][/tex]
