Question: ALGEBRA Find the value of $x$ and $KL$ if $K$ is between $J$ and $L$.
4. $JK = 6x$, $KL = 3x$, and $JL = 27$
5. $JK = 2x$, $KL = x + 2$, and $JL = 5x - 10$
1. **Problem Statement:**
Find the values of $x$ and $KL$ given that point $K$ lies between points $J$ and $L$ on a line segment.
2. **Key Concept:**
If $K$ is between $J$ and $L$, then the segment addition postulate applies:
$$JL = JK + KL$$
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### Problem 4:
Given:
$$JK = 6x, \quad KL = 3x, \quad JL = 27$$
Using the segment addition postulate:
$$JL = JK + KL$$
Substitute values:
$$27 = 6x + 3x$$
$$27 = 9x$$
Divide both sides by 9:
$$\cancel{9}x = \frac{27}{\cancel{9}}$$
$$x = 3$$
Find $KL$:
$$KL = 3x = 3 \times 3 = 9$$
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### Problem 5:
Given:
$$JK = 2x, \quad KL = x + 2, \quad JL = 5x - 10$$
Using the segment addition postulate:
$$JL = JK + KL$$
Substitute values:
$$5x - 10 = 2x + (x + 2)$$
Simplify right side:
$$5x - 10 = 3x + 2$$
Subtract $3x$ from both sides:
$$5x - 3x - 10 = 2$$
$$2x - 10 = 2$$
Add 10 to both sides:
$$2x = 12$$
Divide both sides by 2:
$$\cancel{2}x = \frac{12}{\cancel{2}}$$
$$x = 6$$
Find $KL$:
$$KL = x + 2 = 6 + 2 = 8$$