Alright, let's solve the given equation step-by-step:
[tex]\[
\frac{7}{a+7} = \frac{6}{a+3}
\][/tex]
1. Cross-Multiply to eliminate the fractions:
[tex]\[
7(a + 3) = 6(a + 7)
\][/tex]
2. Expand both sides:
[tex]\[
7a + 21 = 6a + 42
\][/tex]
3. Isolate the variable 'a' by moving the terms involving 'a' to one side and constant terms to the other side. Subtract [tex]\(6a\)[/tex] from both sides:
[tex]\[
7a - 6a + 21 = 42
\][/tex]
This simplifies to:
[tex]\[
a + 21 = 42
\][/tex]
4. Solve for 'a' by subtracting 21 from both sides:
[tex]\[
a = 42 - 21
\][/tex]
Simplifying the right-hand side:
[tex]\[
a = 21
\][/tex]
5. Check the solution to ensure it satisfies the original equation. Substitute [tex]\(a = 21\)[/tex] back into the original equation:
[tex]\[
\frac{7}{21 + 7} = \frac{6}{21 + 3}
\][/tex]
Simplify both sides:
[tex]\[
\frac{7}{28} = \frac{6}{24}
\][/tex]
Both fractions can be reduced:
[tex]\[
\frac{1}{4} = \frac{1}{4}
\][/tex]
Since both sides are equal, the solution is correct.
Therefore, the correct choice is:
[tex]\[
\boxed{a = 21}
\][/tex]
