Certainly! Let's start by considering the original equation:
[tex]\[ a = 180(n - 2) \][/tex]
First, we'll solve this equation for [tex]\( n \)[/tex]:
[tex]\[ a = 180(n - 2) \][/tex]
To isolate [tex]\( n \)[/tex]:
[tex]\[ a = 180n - 360 \][/tex]
Add 360 to both sides:
[tex]\[ a + 360 = 180n \][/tex]
Now, divide both sides by 180:
[tex]\[ n = \frac{a + 360}{180} \][/tex]
So, we have derived the equation:
[tex]\[ n = \frac{a + 360}{180} \][/tex]
Now, let's compare this derived equation with the given equations to check for equivalence:
1. First given equation:
[tex]\[ n = \frac{a}{180} + 1 \][/tex]
The derived equation is:
[tex]\[ n = \frac{a + 360}{180} \][/tex]
To check if these are equivalent, let’s rewrite the derived equation for comparison:
[tex]\[ n = \frac{a}{180} + \frac{360}{180} \][/tex]
Simplifying the fraction:
[tex]\[ n = \frac{a}{180} + 2 \][/tex]
Clearly, this does not match the first given equation which is:
[tex]\[ n = \frac{a}{180} + 1 \][/tex]
Hence, the first equation is not equivalent to the original equation.
2. Second given equation:
[tex]\[ n = \frac{a}{180} + 2 \][/tex]
We have already seen from the above simplification that:
[tex]\[ n = \frac{a + 360}{180} \][/tex]
simplifies to:
[tex]\[ n = \frac{a}{180} + 2 \][/tex]
This exactly matches the second given equation. Hence, the second equation is equivalent to the original equation.
3. Third given equation:
[tex]\[ n = \frac{a + 360}{180} \][/tex]
This is exactly what we derived from the original equation:
[tex]\[ n = \frac{a + 360}{180} \][/tex]
Hence, the third equation is equivalent to the original equation.
Conclusion:
- [tex]\( n = \frac{a}{180} + 1 \)[/tex] is not equivalent to the original equation.
- [tex]\( n = \frac{a}{180} + 2 \)[/tex] is equivalent to the original equation.
- [tex]\( n = \frac{a + 360}{180} \)[/tex] is equivalent to the original equation.
