The problem involves finding the [tex]\( x \)[/tex]-coordinate of point [tex]\( Q \)[/tex] given that point [tex]\( R \)[/tex] divides the line segment [tex]\( \overline{P Q} \)[/tex] in the ratio [tex]\( 1: 3 \)[/tex]. Let's use the section formula to solve this problem step-by-step.
1. Identifying the known values:
- [tex]\( x_R = -1 \)[/tex] (the [tex]\( x \)[/tex]-coordinate of [tex]\( R \)[/tex])
- [tex]\( x_P = -3 \)[/tex] (the [tex]\( x \)[/tex]-coordinate of [tex]\( P \)[/tex])
- The ratio [tex]\( R \)[/tex] divides [tex]\( \overline{P Q} \)[/tex] is [tex]\( 1: 3 \)[/tex]. This means:
- [tex]\( P \)[/tex] to [tex]\( R \)[/tex] ratio ([tex]\( m \)[/tex]) is 3
- [tex]\( R \)[/tex] to [tex]\( Q \)[/tex] ratio ([tex]\( n \)[/tex]) is 1
2. Section formula for finding coordinates:
The section formula states that if a point [tex]\( R \)[/tex] divides a line segment [tex]\( \overline{P Q} \)[/tex] in the ratio [tex]\( m: n \)[/tex], then the coordinates of point [tex]\( R \)[/tex] [tex]\((x_R)\)[/tex] are given by:
[tex]\[
x_R = \frac{{m \cdot x_Q + n \cdot x_P}}{{m + n}}
\][/tex]
3. Setting up the equation using the given values:
Substituting the known values into the section formula:
[tex]\[
-1 = \frac{{3 \cdot x_Q + 1 \cdot (-3)}}{{3 + 1}}
\][/tex]
4. Simplifying the equation:
[tex]\[
-1 = \frac{{3 x_Q - 3}}{4}
\][/tex]
5. Solving for [tex]\( x_Q \)[/tex]:
- First, multiply both sides by 4 to clear the fraction:
[tex]\[
-1 \times 4 = 3 x_Q - 3
\][/tex]
[tex]\[
-4 = 3 x_Q - 3
\][/tex]
- Next, add 3 to both sides to isolate the term involving [tex]\( x_Q \)[/tex]:
[tex]\[
-4 + 3 = 3 x_Q
\][/tex]
[tex]\[
-1 = 3 x_Q
\][/tex]
- Finally, divide both sides by 3 to solve for [tex]\( x_Q \)[/tex]:
[tex]\[
x_Q = \frac{-1}{3}
\][/tex]
[tex]\[
x_Q = -\frac{1}{3}
\][/tex]
Therefore, the [tex]\( x \)[/tex]-coordinate of [tex]\( Q \)[/tex] is [tex]\( -\frac{1}{3} \)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{-\frac{1}{3}}
\][/tex]
