To find the total number of guests at Joy's wedding, we need to set up an equation considering all the meals chosen.
Let's denote the total number of guests by [tex]\( T \)[/tex].
We know the following:
- [tex]\(\frac{1}{3}\)[/tex] of the guests chose beef.
- [tex]\(\frac{5}{12}\)[/tex] of the guests chose chicken.
- 69 guests chose vegetarian.
Given that these are the only meal choices, we can write an equation that sums these fractions to equal the total number of guests [tex]\( T \)[/tex]:
[tex]\[
\frac{1}{3}T + \frac{5}{12}T + 69 = T
\][/tex]
First, we need to combine the fractions [tex]\(\frac{1}{3}\)[/tex] and [tex]\(\frac{5}{12}\)[/tex]. To do this, we find a common denominator for the fractions:
[tex]\[
\frac{1}{3} = \frac{4}{12}
\][/tex]
So the equation becomes:
[tex]\[
\frac{4}{12}T + \frac{5}{12}T + 69 = T
\][/tex]
Combine the fractions on the left side:
[tex]\[
\left(\frac{4}{12} + \frac{5}{12}\right)T + 69 = T
\][/tex]
[tex]\[
\frac{9}{12}T + 69 = T
\][/tex]
Simplify the fraction [tex]\(\frac{9}{12}\)[/tex]:
[tex]\[
\frac{9}{12} = \frac{3}{4}
\][/tex]
Therefore, the equation now is:
[tex]\[
\frac{3}{4}T + 69 = T
\][/tex]
Next, we'll isolate [tex]\( T \)[/tex] by subtracting [tex]\(\frac{3}{4}T\)[/tex] from both sides of the equation:
[tex]\[
69 = T - \frac{3}{4}T
\][/tex]
Simplify the right side:
[tex]\[
69 = \frac{1}{4}T
\][/tex]
To solve for [tex]\( T \)[/tex], multiply both sides by 4:
[tex]\[
T = 69 \times 4
\][/tex]
[tex]\[
T = 276
\][/tex]
Thus, the total number of guests at the wedding is:
[tex]\[
\boxed{276}
\][/tex]
