To determine the equivalence of the given equations, we can solve for [tex]\( a \)[/tex] in the terms of [tex]\( n \)[/tex] and compare the results. Let’s start by examining each equation.
1. Equation 1: [tex]\( n = \frac{a}{180} + 1 \)[/tex]
To solve for [tex]\( a \)[/tex]:
[tex]\[
n = \frac{a}{180} + 1
\][/tex]
Subtract 1 from both sides:
[tex]\[
n - 1 = \frac{a}{180}
\][/tex]
Multiply both sides by 180:
[tex]\[
a = 180(n - 1)
\][/tex]
Simplifying:
[tex]\[
a = 180n - 180
\][/tex]
2. Equation 2: [tex]\( n = \frac{a}{180} + 2 \)[/tex]
To solve for [tex]\( a \)[/tex]:
[tex]\[
n = \frac{a}{180} + 2
\][/tex]
Subtract 2 from both sides:
[tex]\[
n - 2 = \frac{a}{180}
\][/tex]
Multiply both sides by 180:
[tex]\[
a = 180(n - 2)
\][/tex]
Simplifying:
[tex]\[
a = 180n - 360
\][/tex]
3. Equation 3: [tex]\( n = \frac{a + 360}{180} \)[/tex]
To solve for [tex]\( a \)[/tex]:
[tex]\[
n = \frac{a + 360}{180}
\][/tex]
Multiply both sides by 180:
[tex]\[
180n = a + 360
\][/tex]
Subtract 360 from both sides:
[tex]\[
a = 180n - 360
\][/tex]
Now, we compare the expressions for [tex]\( a \)[/tex] obtained from each equation:
- From Equation 1, we got [tex]\( a = 180n - 180 \)[/tex].
- From Equation 2, we got [tex]\( a = 180n - 360 \)[/tex].
- From Equation 3, we also got [tex]\( a = 180n - 360 \)[/tex].
Let's identify the equivalences:
- Equation 2 and Equation 3 yield the same expression: [tex]\( a = 180n - 360 \)[/tex], so they are equivalent.
- Equation 1 yields a different expression ([tex]\( a = 180n - 180 \)[/tex]) compared to Equation 2 and Equation 3.
Therefore, the final result is as follows:
- Equations 2 and 3 are equivalent.
- Equations 1 and 2 are not equivalent.
- Equations 1 and 3 are not equivalent.
So, the summary is:
- [tex]\( a = 180n - 180 \)[/tex] (Equation 1)
- [tex]\( a = 180n - 360 \)[/tex] (Equation 2 and Equation 3)
Equivalence check:
- Equation 1 and Equation 2: Not equivalent.
- Equation 1 and Equation 3: Not equivalent.
- Equation 2 and Equation 3: Equivalent.
