Certainly! Let's solve the equation step-by-step:
[tex]$\sqrt{5a + 2} - \sqrt{7a - 8} = 0$[/tex]
Here's the process to solve it:
1. Isolate one of the square root terms: We start by isolating one of the square root expressions on one side of the equation.
[tex]\[
\sqrt{5a + 2} = \sqrt{7a - 8}
\][/tex]
2. Square both sides: To eliminate the square roots, we square both sides of the equation.
[tex]\[
(\sqrt{5a + 2})^2 = (\sqrt{7a - 8})^2
\][/tex]
This simplifies to:
[tex]\[
5a + 2 = 7a - 8
\][/tex]
3. Solve for [tex]\( a \)[/tex]: Now, let's solve this linear equation for [tex]\( a \)[/tex].
[tex]\[
5a + 2 = 7a - 8
\][/tex]
Subtract [tex]\( 5a \)[/tex] from both sides:
[tex]\[
2 = 2a - 8
\][/tex]
Add 8 to both sides:
[tex]\[
10 = 2a
\][/tex]
Finally, divide both sides by 2:
[tex]\[
a = 5
\][/tex]
4. Verify the solution: It's important to verify that our solution satisfies the original equation.
Plug [tex]\( a = 5 \)[/tex] back into the original equation:
[tex]\[
\sqrt{5(5) + 2} - \sqrt{7(5) - 8} = 0
\][/tex]
Simplify inside the square roots:
[tex]\[
\sqrt{25 + 2} - \sqrt{35 - 8} = 0
\][/tex]
[tex]\[
\sqrt{27} - \sqrt{27} = 0
\][/tex]
[tex]\[
0 = 0
\][/tex]
The solution satisfies the original equation, so [tex]\( a = 5 \)[/tex] is indeed the correct answer.
Thus, the solution to the equation is:
[tex]\[
a = 5
\][/tex]
