Sure, let's solve this step by step.
We are given the points [tex]\( Q \)[/tex] and [tex]\( S \)[/tex] on a number line:
[tex]\[ Q = -14 \][/tex]
[tex]\[ S = 2 \][/tex]
The ratio in which point [tex]\( R \)[/tex] partitions the segment is given as [tex]\( 3:5 \)[/tex], so [tex]\( m = 3 \)[/tex] and [tex]\( n = 5 \)[/tex].
We need to use the formula:
[tex]\[
\left(\frac{m}{m+n}\right) (x_2 - x_1) + x_1
\][/tex]
where:
- [tex]\( x_1 = Q = -14 \)[/tex]
- [tex]\( x_2 = S = 2 \)[/tex]
- [tex]\( m = 3 \)[/tex]
- [tex]\( n = 5 \)[/tex]
Substitute the given values into the formula:
[tex]\[
\left(\frac{3}{3+5}\right) (2 - (-14)) + (-14)
\][/tex]
First, compute the denominator in the fraction:
[tex]\[
3 + 5 = 8
\][/tex]
Next, compute the difference inside the parentheses:
[tex]\[
2 - (-14) = 2 + 14 = 16
\][/tex]
Substitute these values back into the formula:
[tex]\[
\left(\frac{3}{8}\right) \cdot 16 + (-14)
\][/tex]
Now, perform the multiplication:
[tex]\[
\frac{3}{8} \cdot 16 = 6
\][/tex]
Finally, add this value to [tex]\(-14\)[/tex]:
[tex]\[
6 + (-14) = -8
\][/tex]
Thus, the correct expression is:
[tex]\[
\left(\frac{3}{3+5}\right) (2 - (-14)) + (-14)
\][/tex]
So, the answer is:
[tex]\[
\left(\frac{3}{3+5}\right) (2 - (-14)) + (-14)
\][/tex]
Hence, the correct expression using the formula is:
[tex]\[
\left(\frac{3}{3+5}\right) (2-(-14)) + (-14)
\][/tex]
And the expression simplifies to:
[tex]\[
-8
\][/tex]
Therefore, the correct answer from the given options is:
[tex]\[
\left(\frac{3}{3+5}\right)(2-(-14))+(-14)
\][/tex]
