To solve the equation [tex]\(\sqrt{5y + 2} = \sqrt{3y + 14}\)[/tex], we will proceed step-by-step:
1. Square Both Sides:
To eliminate the square roots, square both sides of the equation. This gives us:
[tex]\[
(\sqrt{5y + 2})^2 = (\sqrt{3y + 14})^2
\][/tex]
Simplifying both sides, we get:
[tex]\[
5y + 2 = 3y + 14
\][/tex]
2. Isolate the Variable [tex]\(y\)[/tex]:
To isolate [tex]\(y\)[/tex], we need to move the terms involving [tex]\(y\)[/tex] to one side and constants to the other. First, subtract [tex]\(3y\)[/tex] from both sides:
[tex]\[
5y + 2 - 3y = 3y + 14 - 3y
\][/tex]
Simplifying this, we get:
[tex]\[
2y + 2 = 14
\][/tex]
3. Solve for [tex]\(y\)[/tex]:
Next, subtract 2 from both sides to isolate the term with [tex]\(y\)[/tex]:
[tex]\[
2y + 2 - 2 = 14 - 2
\][/tex]
This simplifies to:
[tex]\[
2y = 12
\][/tex]
Finally, divide both sides by 2 to solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{12}{2}
\][/tex]
[tex]\[
y = 6
\][/tex]
4. Verify the Solution:
To ensure that our solution is correct, substitute [tex]\(y = 6\)[/tex] back into the original equation:
[tex]\[
\sqrt{5(6) + 2} = \sqrt{3(6) + 14}
\][/tex]
Simplifying inside the square roots:
[tex]\[
\sqrt{30 + 2} = \sqrt{18 + 14}
\][/tex]
[tex]\[
\sqrt{32} = \sqrt{32}
\][/tex]
Both sides are equal, confirming our solution.
Therefore, the solution to the equation [tex]\(\sqrt{5y + 2} = \sqrt{3y + 14}\)[/tex] is:
[tex]\[
y = 6
\][/tex]
