Answer :
To determine the [tex]\( x \)[/tex]-coordinate of point [tex]\( Q \)[/tex], let's use the section formula. Given that point [tex]\( R \)[/tex] divides the segment [tex]\( \overline{PQ} \)[/tex] in the ratio [tex]\( 1:3 \)[/tex], and we know the [tex]\( x \)[/tex]-coordinate of [tex]\( R \)[/tex] is -1, and the [tex]\( x \)[/tex]-coordinate of [tex]\( P \)[/tex] is -3, we are to find the [tex]\( x \)[/tex]-coordinate of [tex]\( Q \)[/tex].
The section formula for a point [tex]\( R \)[/tex] that divides the segment [tex]\( \overline{PQ} \)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[ R_x = \frac{m \cdot Q_x + n \cdot P_x}{m + n} \][/tex]
In this problem:
- [tex]\( R_x = -1 \)[/tex]
- [tex]\( P_x = -3 \)[/tex]
- [tex]\( \frac{RQ}{PQ} = \frac{1}{3} \)[/tex] which gives [tex]\( m = 1 \)[/tex] and [tex]\( n = 3 \)[/tex]
Substituting the known values into the formula, we get:
[tex]\[ -1 = \frac{1 \cdot Q_x + 3 \cdot (-3)}{1 + 3} \][/tex]
First, simplify the equation on the right-hand side:
[tex]\[ -1 = \frac{Q_x - 9}{4} \][/tex]
Now, eliminate the fraction by multiplying both sides by 4:
[tex]\[ -1 \cdot 4 = Q_x - 9 \][/tex]
This gives:
[tex]\[ -4 = Q_x - 9 \][/tex]
To solve for [tex]\( Q_x \)[/tex], add 9 to both sides of the equation:
[tex]\[ -4 + 9 = Q_x \][/tex]
[tex]\[ Q_x = 5 \][/tex]
Thus, the [tex]\( x \)[/tex]-coordinate of [tex]\( Q \)[/tex] is [tex]\( 5 \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{5} \][/tex]
The section formula for a point [tex]\( R \)[/tex] that divides the segment [tex]\( \overline{PQ} \)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[ R_x = \frac{m \cdot Q_x + n \cdot P_x}{m + n} \][/tex]
In this problem:
- [tex]\( R_x = -1 \)[/tex]
- [tex]\( P_x = -3 \)[/tex]
- [tex]\( \frac{RQ}{PQ} = \frac{1}{3} \)[/tex] which gives [tex]\( m = 1 \)[/tex] and [tex]\( n = 3 \)[/tex]
Substituting the known values into the formula, we get:
[tex]\[ -1 = \frac{1 \cdot Q_x + 3 \cdot (-3)}{1 + 3} \][/tex]
First, simplify the equation on the right-hand side:
[tex]\[ -1 = \frac{Q_x - 9}{4} \][/tex]
Now, eliminate the fraction by multiplying both sides by 4:
[tex]\[ -1 \cdot 4 = Q_x - 9 \][/tex]
This gives:
[tex]\[ -4 = Q_x - 9 \][/tex]
To solve for [tex]\( Q_x \)[/tex], add 9 to both sides of the equation:
[tex]\[ -4 + 9 = Q_x \][/tex]
[tex]\[ Q_x = 5 \][/tex]
Thus, the [tex]\( x \)[/tex]-coordinate of [tex]\( Q \)[/tex] is [tex]\( 5 \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{5} \][/tex]
