Let's analyze each given equation and determine if they are equivalent to the first equation [tex]\( a = 180 (n - 2) \)[/tex].
First, let's convert the first equation [tex]\( a = 180 (n - 2) \)[/tex] to express [tex]\( n \)[/tex] in terms of [tex]\( a \)[/tex]:
1. [tex]\( a = 180 (n - 2) \)[/tex]
2. Distribute the [tex]\( 180 \)[/tex]: [tex]\( a = 180n - 360 \)[/tex]
3. Add [tex]\( 360 \)[/tex] to both sides: [tex]\( a + 360 = 180n \)[/tex]
4. Divide by [tex]\( 180 \)[/tex]: [tex]\( n = \frac{a + 360}{180} \)[/tex]
So, the equivalent form of the first equation is:
[tex]\[ n = \frac{a + 360}{180} \][/tex]
Next, let's examine the other equations to check for equivalence.
- Second equation: [tex]\( n = \frac{a}{180} + 1 \)[/tex]
Attempt to convert this form back to the original form of the first equation:
[tex]\[ n = \frac{a}{180} + 1 \][/tex]
[tex]\[ a = 180(n - 1) \][/tex]
[tex]\[ a = 180n - 180 \][/tex]
It is not the same as [tex]\( a = 180(n - 2) \)[/tex], thus it is not equivalent.
- Third equation: [tex]\( n = \frac{a}{180} + 2 \)[/tex]
Convert this form:
[tex]\[ n = \frac{a}{180} + 2 \][/tex]
[tex]\[ a = 180(n - 2) \][/tex]
[tex]\[ a = 180n - 360 \][/tex]
This appears to match the form of the first equation. Hence, this is a candidate for equivalence, but we will confirm it separately.
- Fourth equation: [tex]\( n = \frac{a + 360}{180} \)[/tex]
Simply convert this:
[tex]\[ n = \frac{a + 360}{180} \][/tex]
[tex]\[ a = 180n - 360 \][/tex]
This matches exactly with the rearranged form of the first equation. Thus, it is equivalent.
Summarizing the findings:
- The second equation [tex]\( n = \frac{a}{180} + 1 \)[/tex] is not equivalent to [tex]\( a = 180(n - 2) \)[/tex].
- The third equation [tex]\( n = \frac{a}{180} + 2 \)[/tex] initially seemed equivalent, but further analysis confirmed it is not equivalent because it leads to a different form.
- The fourth equation [tex]\( n = \frac{a+360}{180} \)[/tex] is equivalent to [tex]\( a = 180(n-2) \)[/tex].
After confirming the detailed analysis, we conclude none of the other equations are equivalent to the first one, [tex]\( a = 180(n - 2) \)[/tex].
