Let's analyze the problem and apply the formula effectively.
We are given:
- The point [tex]\( Q \)[/tex] with coordinate [tex]\( x_1 = -14 \)[/tex]
- The point [tex]\( S \)[/tex] with coordinate [tex]\( x_2 = 2 \)[/tex]
- The ratio [tex]\( R \)[/tex] partitions the line segment [tex]\( QS \)[/tex] in the ratio [tex]\( 3:5 \)[/tex], thus [tex]\( m = 3 \)[/tex] and [tex]\( n = 5 \)[/tex]
The formula to find the location of point [tex]\( R \)[/tex] is given by:
[tex]\[ \left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1 \][/tex]
Substitute the known values into the formula to confirm whether it correctly finds the location of point [tex]\( R \)[/tex]:
[tex]\[ \left(\frac{3}{3+5}\right)(2 - (-14)) + (-14) \][/tex]
So, the correct expression among the given options is:
[tex]\[ \left(\frac{3}{3+5}\right)(2 - (-14)) + (-14) \][/tex]
Therefore, the correct expression that uses the formula to find the location of point [tex]\( R \)[/tex] is:
[tex]\[ \left(\frac{3}{3+5}\right)(2 - (-14)) + (-14) \][/tex]
Evaluating this expression yields the numerical result, which confirms that point [tex]\( R \)[/tex] is located at [tex]\(-8.0\)[/tex] on the number line.
