Let's solve the problem step by step.
Given the equations:
1. [tex]\( 6a - 8b = 0 \)[/tex]
2. [tex]\( c = 12b \)[/tex]
First, we solve the first equation for [tex]\(a\)[/tex] in terms of [tex]\(b\)[/tex]:
[tex]\[ 6a - 8b = 0 \][/tex]
[tex]\[ 6a = 8b \][/tex]
[tex]\[ a = \frac{8b}{6} \][/tex]
[tex]\[ a = \frac{4b}{3} \][/tex]
Now, we have [tex]\(a\)[/tex] expressed in terms of [tex]\(b\)[/tex]:
[tex]\[ a = \frac{4}{3}b \][/tex]
Next, we need to find the ratio of [tex]\(a\)[/tex] to [tex]\(c\)[/tex]. Using the second given equation:
[tex]\[ c = 12b \][/tex]
We substitute [tex]\(a = \frac{4}{3}b\)[/tex] and [tex]\(c = 12b\)[/tex] into the ratio [tex]\(\frac{a}{c}\)[/tex]:
[tex]\[ \frac{a}{c} = \frac{\frac{4}{3}b}{12b} \][/tex]
Simplify the expression:
[tex]\[ \frac{a}{c} = \frac{\frac{4}{3}b}{12b} \][/tex]
[tex]\[ \frac{a}{c} = \frac{4b}{3 \cdot 12b} \][/tex]
[tex]\[ \frac{a}{c} = \frac{4b}{36b} \][/tex]
[tex]\[ \frac{a}{c} = \frac{4}{36} \][/tex]
[tex]\[ \frac{a}{c} = \frac{1}{9} \][/tex]
Thus, the ratio of [tex]\(a\)[/tex] to [tex]\(c\)[/tex] is [tex]\(1\)[/tex] to [tex]\(9\)[/tex].
Therefore, the ratio of [tex]\(a\)[/tex] to [tex]\(c\)[/tex] is equivalent to [tex]\(1\)[/tex] to [tex]\(9\)[/tex].
