To solve the equation [tex]\(\frac{a-4}{4} = \frac{a+1}{6}\)[/tex], let's follow these steps:
1. Clear the fractions by finding a common denominator:
- The common denominator of 4 and 6 is 12.
2. Multiply both sides of the equation by 12 to eliminate the fractions:
[tex]\[
12 \cdot \frac{a-4}{4} = 12 \cdot \frac{a+1}{6}
\][/tex]
3. Simplify both sides:
- When we multiply [tex]\(\frac{a-4}{4}\)[/tex] by 12, we get:
[tex]\[
12 \cdot \frac{a-4}{4} = 3(a - 4)
\][/tex]
- When we multiply [tex]\(\frac{a+1}{6}\)[/tex] by 12, we get:
[tex]\[
12 \cdot \frac{a+1}{6} = 2(a + 1)
\][/tex]
4. Rewrite the equation without fractions:
[tex]\[
3(a - 4) = 2(a + 1)
\][/tex]
5. Distribute on both sides:
- On the left side:
[tex]\[
3a - 12
\][/tex]
- On the right side:
[tex]\[
2a + 2
\][/tex]
So, the equation becomes:
[tex]\[
3a - 12 = 2a + 2
\][/tex]
6. Isolate the variable [tex]\(a\)[/tex]:
- Subtract [tex]\(2a\)[/tex] from both sides:
[tex]\[
3a - 2a - 12 = 2a - 2a + 2
\][/tex]
- Simplify:
[tex]\[
a - 12 = 2
\][/tex]
- Add 12 to both sides:
[tex]\[
a - 12 + 12 = 2 + 12
\][/tex]
- Simplify:
[tex]\[
a = 14
\][/tex]
7. Check the solution by substituting [tex]\(a = 14\)[/tex] back into the original equation:
[tex]\[
\frac{14 - 4}{4} = \frac{14 + 1}{6}
\][/tex]
- Simplify both sides:
[tex]\[
\frac{10}{4} = \frac{15}{6}
\][/tex]
- Further simplify:
[tex]\[
2.5 = 2.5
\][/tex]
Since both sides of the equation are equal, [tex]\(a = 14\)[/tex] is indeed the correct solution.
Thus, the solution to the equation [tex]\(\frac{a-4}{4} = \frac{a+1}{6}\)[/tex] is:
Choice A. [tex]\(a = 14\)[/tex]
