Question

check rachels steps as she solves the problem. if there is a mistake, where does the first mistake occur? assume the sides that look parallel are parallel.
step 1 : \\( \frac { 6 } { 14 } = \frac { 4.5 } { x } \\)
step 2 : \\( 6 x = 63 \\)
step 3 : \\( x = 10.5 \\)

Explanation:

Step 1: Analyze the proportion setup

The correct proportion should be based on the basic proportionality theorem (Thales' theorem). If the lines are parallel, the ratio of the segments of one side should be equal to the ratio of the segments of the other side. The correct proportion is \(\frac{4.5}{4.5 + x}=\frac{6}{14}\), not \(\frac{6}{14}=\frac{4.5}{x}\) as Rachel wrote in Step 1.

Step 2: Cross - multiply the correct proportion

Cross - multiplying \(\frac{4.5}{4.5 + x}=\frac{6}{14}\) gives \(4.5\times14 = 6\times(4.5 + x)\).

Step 3: Expand and solve

\(63=27 + 6x\). Then \(6x=63 - 27=36\), and \(x = 6\).

Answer:

Rachel made a mistake in Step 1. The first incorrect step is Step 1.