To solve for the [tex]\(x\)[/tex]-coordinate of point [tex]\(Q\)[/tex], let's use the section formula. The section formula states that if a point [tex]\(R\)[/tex] divides a line segment [tex]\(\overline{PQ}\)[/tex] in the ratio [tex]\(m:n\)[/tex], and the coordinates of [tex]\(P\)[/tex] and [tex]\(Q\)[/tex] are [tex]\((x_P, y_P)\)[/tex] and [tex]\((x_Q, y_Q)\)[/tex], respectively, then the coordinates of [tex]\(R\)[/tex] are given by:
[tex]\[
(x_R, y_R) = \left( \frac{mx_Q + nx_P}{m+n}, \frac{my_Q + ny_P}{m+n} \right)
\][/tex]
In this problem, we are given:
- The ratio [tex]\(1 : 3\)[/tex], which means [tex]\(m=1\)[/tex] and [tex]\(n=3\)[/tex],
- The [tex]\(x\)[/tex]-coordinates only (no need to worry about [tex]\(y\)[/tex]-coordinates),
- The [tex]\(x\)[/tex]-coordinate of [tex]\(R\)[/tex] is [tex]\(-1\)[/tex],
- The [tex]\(x\)[/tex]-coordinate of [tex]\(P\)[/tex] is [tex]\(-3\)[/tex].
We need to find the [tex]\(x\)[/tex]-coordinate of [tex]\(Q\)[/tex], denoted as [tex]\(x_Q\)[/tex].
Using the section formula for the [tex]\(x\)[/tex]-coordinate, we have:
[tex]\[
x_R = \frac{m x_Q + n x_P}{m + n}
\][/tex]
Substitute the values into the formula:
[tex]\[
-1 = \frac{1 \cdot x_Q + 3 \cdot (-3)}{1 + 3}
\][/tex]
Simplify the expression:
[tex]\[
-1 = \frac{x_Q - 9}{4}
\][/tex]
Multiply both sides of the equation by 4 to clear the denominator:
[tex]\[
-4 = x_Q - 9
\][/tex]
Solve for [tex]\(x_Q\)[/tex]:
[tex]\[
x_Q = -4 + 9
\][/tex]
[tex]\[
x_Q = 5
\][/tex]
Thus, the [tex]\(x\)[/tex]-coordinate of point [tex]\(Q\)[/tex] is [tex]\(5\)[/tex].
So, the correct answer is:
C. [tex]\(5\)[/tex]
