Let's solve the problem step-by-step using the section formula.
Given data:
- Point [tex]\( R \)[/tex] divides [tex]\( \overline{PQ} \)[/tex] in the ratio [tex]\( 1:3 \)[/tex].
- 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].
The section formula for the [tex]\( x \)[/tex]-coordinate, when a point [tex]\( R(x_R) \)[/tex] divides the line segment [tex]\( \overline{PQ} \)[/tex] in the ratio [tex]\( a:b \)[/tex], is given by:
[tex]\[ x_R = \frac{b \cdot x_P + a \cdot x_Q}{a + b} \][/tex]
Here, [tex]\( a = 1 \)[/tex] and [tex]\( b = 3 \)[/tex]:
[tex]\[ x_R = -1 \][/tex]
[tex]\[ x_P = -3 \][/tex]
Substitute these values into the formula:
[tex]\[ -1 = \frac{3 \cdot (-3) + 1 \cdot x_Q}{1 + 3} \][/tex]
Now, solve for [tex]\( x_Q \)[/tex]:
1. Simplify the expression inside the fraction:
[tex]\[ -1 = \frac{-9 + x_Q}{4} \][/tex]
2. Multiply both sides by 4 to eliminate the fraction:
[tex]\[ -4 = -9 + x_Q \][/tex]
3. Solve for [tex]\( x_Q \)[/tex]:
[tex]\[ x_Q = -4 + 9 \][/tex]
[tex]\[ x_Q = 5 \][/tex]
Therefore, the [tex]\( x \)[/tex]-coordinate of [tex]\( Q \)[/tex] is [tex]\(\boxed{5}\)[/tex].
