To find the value of [tex]\(\frac{a}{b}\)[/tex] given the equation
[tex]\[ \frac{8a + 5b}{8a - 5b} = \frac{7}{3}, \][/tex]
we will follow these steps:
1. Cross-multiply to eliminate the fraction:
[tex]\[ 3 \cdot (8a + 5b) = 7 \cdot (8a - 5b). \][/tex]
2. Distribute the multiplication on both sides:
[tex]\[ 3 \cdot 8a + 3 \cdot 5b = 7 \cdot 8a - 7 \cdot 5b, \][/tex]
which simplifies to:
[tex]\[ 24a + 15b = 56a - 35b. \][/tex]
3. Move all terms involving [tex]\(a\)[/tex] to one side and terms involving [tex]\(b\)[/tex] to the other:
[tex]\[ 24a - 56a = -35b - 15b, \][/tex]
which simplifies to:
[tex]\[ -32a = -50b. \][/tex]
4. Divide both sides by [tex]\(-32\)[/tex] to solve for [tex]\(a\)[/tex] in terms of [tex]\(b\)[/tex]:
[tex]\[ a = \frac{-50}{-32} b. \][/tex]
5. Simplify the fraction [tex]\(\frac{-50}{-32}\)[/tex]:
[tex]\[ a = \frac{50}{32} b \][/tex]
[tex]\[ a = \frac{25}{16} b. \][/tex]
6. From the above relation, we can write:
[tex]\[ \frac{a}{b} = \frac{25}{16}. \][/tex]
Therefore, the value of [tex]\(\frac{a}{b}\)[/tex] is
[tex]\[ 1.5625. \][/tex]
