To solve the equation [tex]\(6(9 y - 1) - 10(5 y) - 3 y = -4(2 y - 12) + 8(y - 6)\)[/tex], follow these steps:
### Step 1: Distribute the constants within the parentheses.
Expand both sides of the equation:
[tex]\[ 6(9 y - 1) = 6 \cdot 9 y - 6 \cdot 1 = 54 y - 6 \][/tex]
[tex]\[ -10(5 y) = -10 \cdot 5 y = -50 y \][/tex]
[tex]\[ -4(2 y - 12) = -4 \cdot 2 y + 4 \cdot 12 = -8 y + 48 \][/tex]
[tex]\[ 8(y - 6) = 8 \cdot y - 8 \cdot 6 = 8 y - 48 \][/tex]
So the equation now looks like:
[tex]\[ 54 y - 6 - 50 y - 3 y = -8 y + 48 + 8 y - 48 \][/tex]
### Step 2: Combine like terms on both sides.
Combine the [tex]\( y \)[/tex] terms and constant terms respectively:
[tex]\[ 54 y - 50 y - 3 y - 6 = -8 y + 8 y + 48 - 48 \][/tex]
[tex]\[ (54 y - 50 y - 3 y) - 6 = 0 \][/tex]
[tex]\[ 1 y - 6 = 0 \][/tex]
### Step 3: Isolate the variable [tex]\( y \)[/tex].
Add 6 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[ y - 6 + 6 = 0 + 6 \][/tex]
[tex]\[ y = 6 \][/tex]
Thus, the value of [tex]\( y \)[/tex] is:
[tex]\[ \boxed{6} \][/tex]
