To solve the equation [tex]\( 6\left(y - \frac{1}{3}\right) = 4(3y - 5) \)[/tex], follow these steps:
1. Begin by expanding both sides of the equation:
[tex]\[ 6 \left(y - \frac{1}{3}\right) = 6y - 6 \cdot \frac{1}{3} \][/tex]
Simplify the right side first:
[tex]\[ = 6y - 2 \][/tex]
Similarly for the right side:
[tex]\[ 4(3y - 5) = 4 \cdot 3y - 4 \cdot 5 \][/tex]
[tex]\[ = 12y - 20 \][/tex]
So the equation now is:
[tex]\[ 6y - 2 = 12y - 20 \][/tex]
2. Rearrange the equation to isolate the terms involving [tex]\( y \)[/tex] on one side:
[tex]\[ 6y - 2 = 12y - 20 \][/tex]
3. Move the [tex]\( y \)[/tex] terms to one side and the constants to the other:
Subtract [tex]\( 6y \)[/tex] from both sides:
[tex]\[ -2 = 12y - 6y - 20 \][/tex]
[tex]\[ -2 = 6y - 20 \][/tex]
4. Add 20 to both sides to isolate the [tex]\( y \)[/tex] terms:
[tex]\[ -2 + 20 = 6y \][/tex]
[tex]\[ 18 = 6y \][/tex]
5. Finally, divide both sides by 6 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{18}{6} \][/tex]
[tex]\[ y = 3 \][/tex]
Hence, the solution to the equation is:
[tex]\[ y = 3 \][/tex]
