This isosceles triangle has two sides of equal length, $$ a $$, that are longer than the length of the base, $$ b $$. The perimeter of the triangle is 15.7 centimeters. The equation $$ 2 a+b=15.7 $$ can be used to find the side lengths. If one of the longer sides is 6.3 centimeters, what is the length of the base?

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To find the length of the base $$ b $$ of the isosceles triangle, follow these steps:

1. **Understand the Given Information:**
   - The triangle has two equal sides, each of length $$ a $$.
   - The base of the triangle has length $$ b $$.
   - The perimeter of the triangle is $$ 15.7 $$ centimeters.
   - One of the equal sides ($$ a $$) is $$ 6.3 $$ centimeters.

2. **Set Up the Perimeter Equation:**
   
   The perimeter $$ P $$ of the triangle is the sum of all its sides:
   $$
   P = a + a + b = 2a + b
   $$
   
   Given that $$ P = 15.7 $$ centimeters, the equation becomes:
   $$
   2a + b = 15.7
   $$

3. **Substitute the Known Value of $$ a $$:**
   
   Substitute $$ a = 6.3 $$ centimeters into the equation:
   $$
   2(6.3) + b = 15.7
   $$
   $$
   12.6 + b = 15.7
   $$

4. **Solve for $$ b $$:**
   
   Subtract $$ 12.6 $$ from both sides of the equation to isolate $$ b $$:
   $$
   b = 15.7 - 12.6
   $$
   $$
   b = 3.1 \text{ centimeters}
   $$

**Answer:**  
The length of the base is $$ 3.1 $$ centimeters.
The base is 3.1 centimeters long.

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