Question

1. if \\( \overline { l m } = 3 x + 5, \overline { m n } = 4 x, \\) and \\( \overline { l n } = 11 x - 7, \\) select all that are true.

\\( \square \\) a. \\( x = 3 \\)
\\( \square \\) b. \\( x = 7 \\)
\\( \square \\) c. \\( l m = 14 \\)
\\( \square \\) d. \\( l m = 28 \\)
\\( \square \\) e. \\( l n = 26 \\)
\\( \square \\) f. \\( l n = 54 \\)

Explanation:

Step1: Use segment - addition postulate

Since \(LM + MN=LN\), substitute \(LM = 3x + 5\), \(MN = 4x\), and \(LN=11x - 7\) into the equation: \((3x + 5)+4x=11x - 7\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \(3x+4x + 5=11x - 7\), so \(7x + 5=11x - 7\).

Step3: Solve for \(x\)

Subtract \(7x\) from both sides: \(5 = 11x-7x - 7\), which simplifies to \(5 = 4x-7\). Then add \(7\) to both sides: \(5 + 7=4x\), so \(12 = 4x\). Divide both sides by \(4\): \(x = 3\).

Step4: Find \(LM\)

Substitute \(x = 3\) into \(LM=3x + 5\). Then \(LM=3\times3 + 5=9 + 5=14\).

Step5: Find \(LN\)

Substitute \(x = 3\) into \(LN=11x - 7\). Then \(LN=11\times3-7=33 - 7=26\).

Answer:

A. \(x = 3\), C. \(LM = 14\), E. \(LN = 26\)