Let's prove algebraically that [tex]\(0.\dot{5} = \frac{5}{9}\)[/tex].
1. Assign the repeating decimal to a variable:
Let [tex]\( x = 0.\dot{5} \)[/tex]
2. Since [tex]\( 0.\dot{5} \)[/tex] is a repeating decimal, we know that:
[tex]\( x = 0.555555\ldots \)[/tex]
3. To eliminate the repeating part, multiply both sides of the equation by 10 to shift the decimal point one place to the right:
[tex]\( 10x = 5.555555\ldots \)[/tex]
4. Now, we have two equations:
[tex]\[ x = 0.555555\ldots \][/tex]
[tex]\[ 10x = 5.555555\ldots \][/tex]
5. Subtract the first equation from the second equation:
[tex]\[ 10x - x = 5.555555\ldots - 0.555555\ldots \][/tex]
6. Simplify both sides:
[tex]\[ 9x = 5 \][/tex]
7. Solve for [tex]\( x \)[/tex] by dividing both sides of the equation by 9:
[tex]\[ x = \frac{5}{9} \][/tex]
Therefore, [tex]\( x = \frac{5}{9} \)[/tex].
