Question

1. \\(sqrt{5y - 15} = y - 3\\)\
2. \\(sqrt{41 - 8a} + 7 = a\\)

Explanation:

Step1: Isolate radical (Q13)

$\sqrt{5y-15}=y-3$
First, note domain: $5y-15\geq0 \implies y\geq3$, and $y-3\geq0 \implies y\geq3$

Step2: Square both sides (Q13)

$(\sqrt{5y-15})^2=(y-3)^2$
$5y-15=y^2-6y+9$

Step3: Rearrange to quadratic (Q13)

$y^2-11y+24=0$

Step4: Factor quadratic (Q13)

$(y-3)(y-8)=0$
Solutions: $y=3, y=8$

Step5: Isolate radical (Q15)

$\sqrt{41-8a}=a-7$
Domain: $41-8a\geq0 \implies a\leq\frac{41}{8}=5.125$, and $a-7\geq0 \implies a\geq7$
No overlap in domain initially, but proceed to check.

Step6: Square both sides (Q15)

$(\sqrt{41-8a})^2=(a-7)^2$
$41-8a=a^2-14a+49$

Step7: Rearrange to quadratic (Q15)

$a^2-6a+8=0$

Step8: Factor quadratic (Q15)

$(a-2)(a-4)=0$
Solutions: $a=2, a=4$
Check domain: $2<7, 4<7$, so both violate $a\geq7$. No valid solutions.

Answer:

For question 13: $y=3$ and $y=8$
For question 15: No valid solutions