Certainly! Let's solve each equation step-by-step.
a) [tex]\( 8 + y = 25 \)[/tex]
To find [tex]\( y \)[/tex], subtract 8 from both sides:
[tex]\[
y = 25 - 8
\][/tex]
[tex]\[
y = 17
\][/tex]
b) [tex]\( 2x = 82 \)[/tex]
To find [tex]\( x \)[/tex], divide both sides by 2:
[tex]\[
x = \frac{82}{2}
\][/tex]
[tex]\[
x = 41
\][/tex]
c) [tex]\( 3m = -48 \)[/tex]
To find [tex]\( m \)[/tex], divide both sides by 3:
[tex]\[
m = \frac{-48}{3}
\][/tex]
[tex]\[
m = -16
\][/tex]
d) [tex]\( \frac{x}{2} = 15 \)[/tex]
To find [tex]\( x \)[/tex], multiply both sides by 2:
[tex]\[
x = 15 \times 2
\][/tex]
[tex]\[
x = 30
\][/tex]
e) [tex]\( 27 + p = 43 \)[/tex]
To find [tex]\( p \)[/tex], subtract 27 from both sides:
[tex]\[
p = 43 - 27
\][/tex]
[tex]\[
p = 16
\][/tex]
f) [tex]\( n^2 = 25 \)[/tex]
To find [tex]\( n \)[/tex], take the square root of both sides. Note that there are two solutions since both positive and negative roots are possible:
[tex]\[
n = \sqrt{25} \quad \text{or} \quad n = -\sqrt{25}
\][/tex]
[tex]\[
n = 5 \quad \text{or} \quad n = -5
\][/tex]
g) [tex]\( -\sqrt[3]{x} = -2 \)[/tex]
To find [tex]\( x \)[/tex], first remove the negative sign from both sides:
[tex]\[
\sqrt[3]{x} = 2
\][/tex]
Now cube both sides:
[tex]\[
x = 2^3
\][/tex]
[tex]\[
x = 8
\][/tex]
Since we removed a negative initially, we return [tex]\( x = -8 \)[/tex].
So, the solutions are:
[tex]\[
y = 17, \quad x = 41, \quad m = -16, \quad x = 30, \quad p = 16, \quad n = 5 \text{ or } -5, \quad x = -8
\][/tex]
