To solve for [tex]\( s \)[/tex] and [tex]\( t \)[/tex] given the equations [tex]\(\frac{3}{s} = 7\)[/tex] and [tex]\(\frac{4}{t} = 12\)[/tex], follow these steps:
1. Solve for [tex]\( s \)[/tex] in the equation [tex]\(\frac{3}{s} = 7\)[/tex]:
[tex]\[
\frac{3}{s} = 7
\][/tex]
To isolate [tex]\( s \)[/tex], multiply both sides by [tex]\( s \)[/tex]:
[tex]\[
3 = 7s
\][/tex]
Now, divide both sides by 7:
[tex]\[
s = \frac{3}{7}
\][/tex]
2. Solve for [tex]\( t \)[/tex] in the equation [tex]\(\frac{4}{t} = 12\)[/tex]:
[tex]\[
\frac{4}{t} = 12
\][/tex]
To isolate [tex]\( t \)[/tex], multiply both sides by [tex]\( t \)[/tex]:
[tex]\[
4 = 12t
\][/tex]
Now, divide both sides by 12:
[tex]\[
t = \frac{4}{12} = \frac{1}{3}
\][/tex]
3. Calculate [tex]\( s - t \)[/tex]:
[tex]\[
s = \frac{3}{7}, \quad t = \frac{1}{3}
\][/tex]
To subtract these fractions, find a common denominator. The least common multiple of 7 and 3 is 21. Convert the fractions:
[tex]\[
s = \frac{3}{7} = \frac{3 \times 3}{7 \times 3} = \frac{9}{21}
\][/tex]
[tex]\[
t = \frac{1}{3} = \frac{1 \times 7}{3 \times 7} = \frac{7}{21}
\][/tex]
Subtract the two fractions:
[tex]\[
s - t = \frac{9}{21} - \frac{7}{21} = \frac{2}{21}
\][/tex]
Thus, the value of [tex]\( s - t \)[/tex] is:
[tex]\[
\boxed{\frac{2}{21}}
\][/tex]
