Sure, let's solve each equation step-by-step.
1. Solve for [tex]\( r \)[/tex]:
[tex]\[
11r + 60 = 16
\][/tex]
Start by isolating [tex]\( r \)[/tex]:
[tex]\[
11r = 16 - 60
\][/tex]
Simplify the right-hand side:
[tex]\[
11r = -44
\][/tex]
Divide both sides by 11:
[tex]\[
r = \frac{-44}{11}
\][/tex]
[tex]\[
r = -4
\][/tex]
2. Solve for [tex]\( y \)[/tex]:
[tex]\[
y - 24 = -7
\][/tex]
Add 24 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = -7 + 24
\][/tex]
Simplify the right-hand side:
[tex]\[
y = 17
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
23 - x = 13
\][/tex]
Subtract 23 from both sides:
[tex]\[
-x = 13 - 23
\][/tex]
Simplify the right-hand side:
[tex]\[
-x = -10
\][/tex]
Multiply both sides by -1:
[tex]\[
x = 10
\][/tex]
4. Solve for [tex]\( x \)[/tex] in the equation [tex]\( -67 = 6x - 1 \)[/tex]:
[tex]\[
-67 = 6x - 1
\][/tex]
Add 1 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
-67 + 1 = 6x
\][/tex]
Simplify the left-hand side:
[tex]\[
-66 = 6x
\][/tex]
Divide both sides by 6:
[tex]\[
x = \frac{-66}{6}
\][/tex]
[tex]\[
x = -11
\][/tex]
5. Solve for [tex]\( e \)[/tex]:
[tex]\[
-4e - 9 = 19
\][/tex]
Add 9 to both sides to isolate the term with [tex]\( e \)[/tex]:
[tex]\[
-4e = 19 + 9
\][/tex]
Simplify the right-hand side:
[tex]\[
-4e = 28
\][/tex]
Divide both sides by -4:
[tex]\[
e = \frac{28}{-4}
\][/tex]
[tex]\[
e = -7
\][/tex]
6. Solve for [tex]\( q \)[/tex]:
[tex]\[
-8 = 32 - 5q
\][/tex]
Subtract 32 from both sides:
[tex]\[
-8 - 32 = -5q
\][/tex]
Simplify the left-hand side:
[tex]\[
-40 = -5q
\][/tex]
Divide both sides by -5:
[tex]\[
q = \frac{-40}{-5}
\][/tex]
[tex]\[
q = 8
\][/tex]
7. Solve for [tex]\( k \)[/tex]:
[tex]\[
6 + 10k = 256
\][/tex]
Subtract 6 from both sides to isolate the term with [tex]\( k \)[/tex]:
[tex]\[
10k = 256 - 6
\][/tex]
Simplify the right-hand side:
[tex]\[
10k = 250
\][/tex]
Divide both sides by 10:
[tex]\[
k = \frac{250}{10}
\][/tex]
[tex]\[
k = 25
\][/tex]
So, the solutions are:
[tex]\[
r = -4, \quad y = 17, \quad x = 10, \quad x_1 = -11, \quad e = -7, \quad q = 8, \quad k = 25
\][/tex]
