Point [tex]$J$[/tex] is on line segment [tex]$\overline{IK}$[/tex]. Given [tex]$JK = 2x - 1$[/tex], [tex]$IK = 3x + 2$[/tex], and [tex]$IJ = 3x - 5$[/tex], determine the numerical length of [tex]$\overline{JK}$[/tex].
We know from the given information that point [tex]\(J\)[/tex] is on the line segment [tex]\(\overline{IK}\)[/tex]. This implies that the lengths of the segments [tex]\(JK\)[/tex] and [tex]\(IJ\)[/tex] should sum up to the total length of [tex]\(IK\)[/tex]: [tex]\[
JK + IJ = IK
\][/tex]
2. Set up the equation: Write this relationship as an equation using the given expressions for the segments: [tex]\[
(2x - 1) + (3x - 5) = 3x + 2
\][/tex]
3. Combine like terms: Let's combine like terms on the left-hand side: [tex]\[
2x - 1 + 3x - 5 = 3x + 2
\][/tex] [tex]\[
5x - 6 = 3x + 2
\][/tex]
4. Solve for [tex]\(x\)[/tex]: Isolate [tex]\(x\)[/tex] by first subtracting [tex]\(3x\)[/tex] from both sides: [tex]\[
5x - 3x - 6 = 2
\][/tex] [tex]\[
2x - 6 = 2
\][/tex] Add 6 to both sides to further isolate [tex]\(2x\)[/tex]: [tex]\[
2x = 8
\][/tex] Finally, divide by 2: [tex]\[
x = 4
\][/tex]
5. Find the length of [tex]\(JK\)[/tex]: Substitute [tex]\(x = 4\)[/tex] back into the expression for [tex]\(JK\)[/tex]: [tex]\[
JK = 2x - 1
\][/tex] [tex]\[
JK = 2(4) - 1 = 8 - 1 = 7
\][/tex]
Therefore, the length of [tex]\(\overline{JK}\)[/tex] is [tex]\(\boxed{7}\)[/tex].
