Certainly! Let's solve this problem step-by-step.
The x-coordinates of the two points, [tex]\(J\)[/tex] and [tex]\(K\)[/tex], are given as [tex]\(x_1 = -2\)[/tex] and [tex]\(x_2 = 4\)[/tex], respectively.
We are given the ratio [tex]\(m:n = 2:5\)[/tex]. This means [tex]\(m = 2\)[/tex] and [tex]\(n = 5\)[/tex].
To find the x-coordinate of the point that divides the line segment from [tex]\(J\)[/tex] to [tex]\(K\)[/tex] in the ratio 2:5, we use the following formula:
[tex]\[
x = \left(\frac{m}{m+n}\right) \left(x_2 - x_1\right) + x_1
\][/tex]
1. Plugging in the values [tex]\(m = 2\)[/tex], [tex]\(n = 5\)[/tex], [tex]\(x_1 = -2\)[/tex], and [tex]\(x_2 = 4\)[/tex] into the formula, we get:
[tex]\[
x = \left(\frac{2}{2+5}\right) \left(4 - (-2)\right) + (-2)
\][/tex]
2. Simplify the denominator:
[tex]\[
x = \left(\frac{2}{7}\right) \left(4 + 2\right) + (-2)
\][/tex]
3. Simplify the expression inside the parentheses:
[tex]\[
x = \left(\frac{2}{7}\right) \times 6 + (-2)
\][/tex]
4. Perform the multiplication:
[tex]\[
x = \frac{12}{7} - 2
\][/tex]
5. Convert 2 to a fraction with denominator 7 for easier subtraction:
[tex]\[
x = \frac{12}{7} - \frac{14}{7}
\][/tex]
6. Subtract the fractions:
[tex]\[
x = \frac{12 - 14}{7} = \frac{-2}{7}
\][/tex]
Expressing [tex]\(\frac{-2}{7}\)[/tex] as a decimal, we get approximately:
[tex]\[
x \approx -0.285714
\][/tex]
Thus, the x-coordinate of the point that divides the directed line segment from [tex]\(J\)[/tex] to [tex]\(K\)[/tex] into a ratio of 2:5 is [tex]\(-0.2857142857142858\)[/tex].
