Let's go through the problems one by one and solve them:

7) Solve for $y$:

$3y = 14 + 10y$

• Subtract $10y$ from both sides: $3y - 10y = 14$ $-7y = 14$
• Divide both sides by $-7$: $y = -2$

8) Solve for $x$:

$\frac{1}{2}(18x - 48) = 3(x - 8) + 6x$

• Distribute the $\frac{1}{2}$ on the left side: $9x - 24 = 3(x - 8) + 6x$
• Expand the right side: $9x - 24 = 3x - 24 + 6x$
• Combine like terms: $9x - 24 = 9x - 24$
• This simplifies to $0 = 0$, meaning this equation has infinitely many solutions (it's an identity).

9) Solve for $w$:

$-18 - 6w = 6(1 + 3w)$

• Distribute the 6 on the right side: $-18 - 6w = 6 + 18w$
• Add $6w$ to both sides: $-18 = 6 + 24w$
• Subtract 6 from both sides: $-24 = 24w$
• Divide both sides by 24: $w = -1$

10) Solve for $x$:

$-2(1 - 7x) = 5x + 34$

• Distribute the $-2$: $-2 + 14x = 5x + 34$
• Subtract $5x$ from both sides: $-2 + 9x = 34$
• Add 2 to both sides: $9x = 36$
• Divide by 9: $x = 4$

11) Solve for $m$:

$3(m - 3) = 2(6 - 2m)$

• Distribute on both sides: $3m - 9 = 12 - 4m$
• Add $4m$ to both sides: $7m - 9 = 12$
• Add 9 to both sides: $7m = 21$
• Divide by 7: $m = 3$

12) Solve for $x$:

$-3(4y + 3) + 4(6y + 1) = 43$

• Distribute the constants: $-12y - 9 + 24y + 4 = 43$
• Combine like terms: $12y - 5 = 43$
• Add 5 to both sides: $12y = 48$
• Divide by 12: $y = 4$

13a) Describe and correct the error in solving the equation:

Given: $3x - 7 = -2x + 8$ They added $2x$ to both sides incorrectly, making: $3x + (-2x) = 8 + 7$ This is wrong because the 7 should have been added to the other side: $3x - 7 + 7 = -2x + 8 + 7 \implies 3x = -2x + 15$ Then: $3x + 2x = 15 \implies 5x = 15 \implies x = 3$

13b) Describe and correct the error in solving the equation:

Given: $2(v - 5) = - (3v + 5)$ The mistake here is in combining terms. After distributing: $2v - 10 = -3v - 5$ They added incorrectly. The proper steps should be: $2v + 3v = -5 + 10 \implies 5v = 5 \implies v = 1$

Would you like more details or have any questions? Here are 5 related questions:

1. What happens if you get an identity like in problem 8? How do you interpret the result?
2. Can an equation have no solution? How would you recognize that?
3. What other methods can be used to solve linear equations besides simplification?
4. How do you check your solution after solving an equation?
5. What is the difference between solving an equation with variables on both sides versus a one-sided equation?

Tip: Always verify your answer by plugging it back into the original equation to avoid common mistakes like sign errors!