Jessica made her first error in Step 5. Here's a detailed explanation of each step and why Step 5 is incorrect:
- Step 1: [tex]\((\sqrt{a-9})^2 = (4)^2\)[/tex]
Justification: Square both sides of the equation to eliminate the square root. This step is valid.
- Step 2: [tex]\(a - 9 = 16\)[/tex]
Justification: Simplify the expression after squaring both sides. This step is valid.
- Step 3: [tex]\(a - 9 + 9 = 16 + 9\)[/tex]
Justification: Add 9 to both sides to isolate [tex]\(a\)[/tex]. This step is valid.
- Step 4: [tex]\(a = 25\)[/tex]
Justification: Simplify the equation to find the value of [tex]\(a\)[/tex]. This step is valid.
- Step 5: [tex]\(25 - 9 \neq 4\)[/tex]
Justification: This step is incorrect. When [tex]\(a = 25\)[/tex], we need to substitute back into the original equation to check for an extraneous solution. Substituting [tex]\(a = 25\)[/tex] into the original equation [tex]\(\sqrt{a-9}\)[/tex]:
[tex]\[
\sqrt{25 - 9} = \sqrt{16} = 4
\][/tex]
Since [tex]\(4 = 4\)[/tex], there is no extraneous solution, and [tex]\(a = 25\)[/tex] is correct. Therefore, her conclusion in Step 5 is incorrect.
Thus, Jessica made her first error in Step 5.
