Certainly! Let's solve each equation step-by-step.
1) [tex]\( 4n - 2n = 4 \)[/tex]
Combine like terms:
[tex]\[ (4n - 2n) = 4 \][/tex]
[tex]\[ 2n = 4 \][/tex]
Divide [tex]\( 2n \)[/tex] by 2:
[tex]\[ n = \frac{4}{2} \][/tex]
[tex]\[ n = 2 \][/tex]
2) [tex]\( -12 = 2 + 5v + 2v \)[/tex]
Combine like terms:
[tex]\[ -12 = 2 + 7v \][/tex]
Subtract 2 from both sides:
[tex]\[ -12 - 2 = 7v \][/tex]
[tex]\[ -14 = 7v \][/tex]
Divide both sides by 7:
[tex]\[ v = \frac{-14}{7} \][/tex]
[tex]\[ v = -2 \][/tex]
3) [tex]\( 3 = x + 3 - 5x \)[/tex]
Combine like terms:
[tex]\[ 3 = 3 - 4x \][/tex]
Subtract 3 from both sides:
[tex]\[ 3 - 3 = -4x \][/tex]
[tex]\[ 0 = -4x \][/tex]
Divide by -4:
[tex]\[ x = \frac{0}{-4} \][/tex]
[tex]\[ x = 0 \][/tex]
4) [tex]\( x + 3 - 3 = -6 \)[/tex]
Combine like terms on the left:
[tex]\[ x = -6 \][/tex]
5) [tex]\( -12 = 3 - 2k - 3k \)[/tex]
Combine like terms:
[tex]\[ -12 = 3 - 5k \][/tex]
Subtract 3 from both sides:
[tex]\[ -12 - 3 = -5k \][/tex]
[tex]\[ -15 = -5k \][/tex]
Divide both sides by -5:
[tex]\[ k = \frac{-15}{-5} \][/tex]
[tex]\[ k = 3 \][/tex]
6) [tex]\( -1 = -3r + 2r \)[/tex]
Combine like terms:
[tex]\[ -1 = -r \][/tex]
Divide both sides by -1:
[tex]\[ r = \frac{-1}{-1} \][/tex]
[tex]\[ r = 1 \][/tex]
7) [tex]\( 6 = -3(x + 2) \)[/tex]
Distribute -3:
[tex]\[ 6 = -3x - 6 \][/tex]
Add 6 to both sides:
[tex]\[ 6 + 6 = -3x \][/tex]
[tex]\[ 12 = -3x \][/tex]
Divide both sides by -3:
[tex]\[ x = \frac{12}{-3} \][/tex]
[tex]\[ x = -4 \][/tex]
8) [tex]\( -3(4r - 8) = -36 \)[/tex]
Distribute -3:
[tex]\[ -12r + 24 = -36 \][/tex]
Subtract 24 from both sides:
[tex]\[ -12r = -36 - 24 \][/tex]
[tex]\[ -12r = -60 \][/tex]
Divide both sides by -12:
[tex]\[ r = \frac{-60}{-12} \][/tex]
[tex]\[ r = 5 \][/tex]
9) [tex]\( 24 = 6(-x - 3) \)[/tex]
Distribute 6:
[tex]\[ 24 = -6x - 18 \][/tex]
Add 18 to both sides:
[tex]\[ 24 + 18 = -6x \][/tex]
[tex]\[ 42 = -6x \][/tex]
Divide both sides by -6:
[tex]\[ x = \frac{42}{-6} \][/tex]
[tex]\[ x = -7 \][/tex]
10) [tex]\( 75 = 3(-6n - 5) \)[/tex]
Distribute 3:
[tex]\[ 75 = -18n - 15 \][/tex]
Add 15 to both sides:
[tex]\[ 75 + 15 = -18n \][/tex]
[tex]\[ 90 = -18n \][/tex]
Divide both sides by -18:
[tex]\[ n = \frac{90}{-18} \][/tex]
[tex]\[ n = -5 \][/tex]
Thus, the solutions are:
1) [tex]\( n = 2 \)[/tex]
2) [tex]\( v = -2 \)[/tex]
3) [tex]\( x = 0 \)[/tex]
4) [tex]\( x = -6 \)[/tex]
5) [tex]\( k = 3 \)[/tex]
6) [tex]\( r = 1 \)[/tex]
7) [tex]\( x = -4 \)[/tex]
8) [tex]\( r = 5 \)[/tex]
9) [tex]\( x = -7 \)[/tex]
10) [tex]\( n = -5 \)[/tex]
