To find the [tex]\( x \)[/tex]-coordinate of the point that divides the directed line segment from point [tex]\( J \)[/tex] to point [tex]\( K \)[/tex] into a ratio of [tex]\( 2:5 \)[/tex], you can use the section formula. Let's break it down step-by-step.
1. Identify Coordinates and Ratio:
- Let's denote the coordinates with [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex].
- Starting point [tex]\( J \)[/tex]: [tex]\( x_1 = -4 \)[/tex]
- Ending point [tex]\( K \)[/tex]: [tex]\( x_2 = 4 \)[/tex]
- The given ratio is [tex]\( m:n = 2:5 \)[/tex].
- So, [tex]\( m = 2 \)[/tex]
- And [tex]\( n = 5 \)[/tex]
2. Write the Section Formula:
The section formula gives the coordinate of the point dividing the line segment joining [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the ratio [tex]\( m:n \)[/tex]. For the [tex]\( x \)[/tex]-coordinate, the formula is:
[tex]\[
x = \left( \frac{m}{m+n} \right) (x_2 - x_1) + x_1
\][/tex]
3. Substitute the Values into the Formula:
[tex]\[
x = \left( \frac{2}{2+5} \right) (4 - (-4)) + (-4)
\][/tex]
4. Simplify the Expression:
[tex]\[
x = \left( \frac{2}{7} \right) (4 + 4) - 4
\][/tex]
[tex]\[
x = \left( \frac{2}{7} \right) (8) - 4
\][/tex]
[tex]\[
x = \frac{16}{7} - 4
\][/tex]
5. Subtract the Values:
Convert [tex]\( 4 \)[/tex] to a fraction with denominator [tex]\( 7 \)[/tex] for easier subtraction:
[tex]\[
4 = \frac{28}{7}
\][/tex]
Now subtract:
[tex]\[
x = \frac{16}{7} - \frac{28}{7}
\][/tex]
[tex]\[
x = \frac{16 - 28}{7}
\][/tex]
[tex]\[
x = \frac{-12}{7}
\][/tex]
Convert it back to decimal:
[tex]\[
x \approx -1.7142857142857144
\][/tex]
Therefore, the [tex]\( x \)[/tex]-coordinate of the point that divides the directed line segment from [tex]\( J \)[/tex] to [tex]\( K \)[/tex] in the ratio [tex]\( 2:5 \)[/tex] is approximately [tex]\( -1.7142857142857144 \)[/tex].
