To determine which expression correctly uses the formula [tex]\(\left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1\)[/tex] to find the location of point [tex]\( R \)[/tex], let’s revisit the problem details and align them with the formula variables.
Given:
- Points [tex]\( Q \)[/tex] and [tex]\( S \)[/tex] are at [tex]\(-8\)[/tex] and [tex]\(12\)[/tex] respectively on the number line.
- The segment [tex]\( QS \)[/tex] is partitioned by point [tex]\( R \)[/tex] in a [tex]\(4:1\)[/tex] ratio.
In the formula:
- [tex]\( m \)[/tex] and [tex]\( n \)[/tex] represent the ratio in which the segment is divided. Here, [tex]\( m = 4 \)[/tex] and [tex]\( n = 1 \)[/tex].
- [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex] are the coordinates of points [tex]\( Q \)[/tex] and [tex]\( S \)[/tex] respectively. Thus, [tex]\( x_1 = -8 \)[/tex] and [tex]\( x_2 = 12 \)[/tex].
Now, insert these values into the formula:
[tex]\[
\left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1
= \left(\frac{4}{4+1}\right)(12 - (-8)) + (-8)
\][/tex]
Simplifying the expression:
1. Calculate the sum in the denominator: [tex]\( 4 + 1 = 5 \)[/tex].
2. Compute the difference: [tex]\( 12 - (-8) = 12 + 8 = 20 \)[/tex].
3. Calculate the ratio: [tex]\( \frac{4}{5} \)[/tex].
4. Multiply by the difference: [tex]\( \frac{4}{5} \times 20 = 16 \)[/tex].
5. Add this product to [tex]\( x_1 \)[/tex]: [tex]\( 16 + (-8) = 8 \)[/tex].
Therefore, the correct expression is:
[tex]\[
\left( \frac{4}{4+1} \right)(12 - (-8)) + (-8)
\][/tex]
