Question

after finding that y = 5, sandra is solving for x. she sets up her problem and shows her work. in which step does her first error occur, if any? assume the sides that look parallel are parallel. step 1: \\( \frac { 12 } { 3 + ( x + 1 ) } = \frac { 5 } { 3 } \\) step 2: \\( \frac { 12 } { x + 4 } = \frac { 5 } { 3 } \\) step 3: \\( 36 = 5 ( x + 4 ) \\) step 4: \\( 36 = 5 x + 20 \\) step 5: \\( 16 = 5 x \\) \\( x = \frac { 16 } { 5 } \\)

Explanation:

Step 1: Analyze the proportion setup

By the Basic Proportionality Theorem (Thales' Theorem), if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. The correct proportion should be \(\frac{12}{y}=\frac{(x + 1)}{3}\) and \(\frac{y}{2}=\frac{(x-1)}{1}\) (assuming \(y\) is the segment from the intersection point to the base). But Sandra's Step 1: \(\frac{12}{y}=\frac{y+(x + 1)}{3}\) is wrong. The numerator on the right - hand side should not be \(y+(x + 1)\). It should be just \(x + 1\) because the segments are divided proportionally, not summed in that way.

Answer:

Step 1.