Given $P = 12n - 35$ and $P = 49$, we need to find the number of items sold $n$. Substituting: $49 = 12n - 35$. Add 35 to both sides: $49 + 35 = 12n$, which gives us $84 = 12n$. Divide both sides by 12: $n = 84 ÷ 12 = 7$. They sold 7 items to make a $49 profit. A common error is to subtract 35 from 49 instead of adding it when isolating the variable term. When the constant term is negative in the original equation, you add its absolute value to isolate the variable term.

Given $C = 2.5w + 4$ and $C = 19$, we need to find the weight $w$. Substituting 19 for C: $19 = 2.5w + 4$. Subtract 4 from both sides: $15 = 2.5w$. Divide both sides by 2.5: $w = 15 ÷ 2.5 = 6$. The package weighs 6 pounds. A common mistake is to divide 19 by 2.5 without first subtracting the base fee of $4. Remember that in linear cost models, there's often a fixed component that must be accounted for before finding the variable rate.

We need to solve $\frac{3}{4}(x-8) + 5 = 17$ for the original price $x$. First, subtract 5 from both sides: $\frac{3}{4}(x-8) = 12$. Next, multiply both sides by $\frac{4}{3}$ to eliminate the fraction: $x - 8 = 12 × \frac{4}{3} = 16$. Finally, add 8 to both sides: $x = 16 + 8 = 24$. The original price was $24. A common error is to distribute $\frac{3}{4}$ before isolating the parentheses, which creates unnecessary fractions. When a fraction multiplies an entire expression, isolate that expression first before clearing the fraction.

We need to solve $0.5x + 7 = 0.2x + 16$ to find when the two phone plans cost the same. First, we collect like terms by subtracting $0.2x$ from both sides: $0.5x - 0.2x + 7 = 16$, which simplifies to $0.3x + 7 = 16$. Next, subtract 7 from both sides: $0.3x = 9$. Finally, divide by 0.3: $x = 9 ÷ 0.3 = 30$. The plans are equal at 30 gigabytes. A common mistake is incorrectly combining the decimal coefficients. To avoid decimal division errors, you can multiply the entire equation by 10 first to work with whole numbers.

We need to solve $12 + 0.08x = 5 + 0.12x$ to find when two phone plans cost the same. First, we move all x-terms to one side by subtracting 0.08x from both sides: $12 = 5 + 0.12x - 0.08x = 5 + 0.04x$. Next, subtract 5 from both sides to get $7 = 0.04x$, then divide by 0.04 to find $x = 7 \div 0.04 = 175$. A common error is getting the wrong sign when moving x-terms, leading to $-0.04x$ and thus $x = -175$. Always double-check your work by substituting back: $12 + 0.08(175) = 12 + 14 = 26$, and $5 + 0.12(175) = 5 + 21 = 26$ ✓.

We need to solve $2.5x - 7.5 = 0.5x + 12.5$ by first moving all x-terms to one side. Subtracting 0.5x from both sides gives $2x - 7.5 = 12.5$. Adding 7.5 to both sides yields $2x = 20$, so $x = 10$. A common mistake is making an arithmetic error when adding 7.5 + 12.5, perhaps getting 19 instead of 20, which would give $x = 9.5$. When working with decimals, take extra care with arithmetic and consider converting to fractions if that helps avoid errors.

We need to solve $d = vt + 12$ for t in terms of d and v. First, subtract 12 from both sides to get $d - 12 = vt$. Then divide both sides by v to isolate t: $t = \frac{d - 12}{v}$. A common error is to divide by v before subtracting 12, which would give $\frac{d}{v} = t + \frac{12}{v}$, leading to $t = \frac{d}{v} - \frac{12}{v}$, or to invert the fraction. When solving for a variable that's being multiplied, first isolate the product, then divide.

We need to solve $\frac{5}{9}(x - 32) = 10$ by isolating x. First, multiply both sides by $\frac{9}{5}$ to clear the fraction: $x - 32 = 10 \times \frac{9}{5} = \frac{90}{5} = 18$. Then add 32 to both sides to get $x = 18 + 32 = 50$. A common mistake is to multiply by $\frac{5}{9}$ instead of its reciprocal $\frac{9}{5}$, which would give $x - 32 = 10 \times \frac{5}{9} = \frac{50}{9} \approx 5.56$, leading to $x \approx 37.56$. When a fraction multiplies a parenthetical expression, divide both sides by that fraction (multiply by its reciprocal) to isolate the parentheses.

We need to solve $x/4 + x/6 = 10$ by finding a common denominator for the fractions. The LCD of 4 and 6 is 12, so we rewrite: $(3x/12) + (2x/12) = 10$, which gives us $5x/12 = 10$. Multiplying both sides by 12 yields $5x = 120$, so $x = 24$. A common error is to add denominators directly (getting $x/10 = 10$, thus $x = 100$) or to use 24 as the LCD instead of 12. When adding fractions, always find the least common denominator first.

To solve $\frac{x-4}{3} + \frac{x+2}{6} = 5$, we need a common denominator of 6. Multiply the first fraction by $\frac{2}{2}$: $\frac{2(x-4)}{6} + \frac{x+2}{6} = 5$. This gives us $\frac{2x-8+x+2}{6} = 5$, which simplifies to $\frac{3x-6}{6} = 5$. Multiply both sides by 6: $3x - 6 = 30$. Add 6 to both sides: $3x = 36$. Divide by 3: $x = 12$. The common mistake is incorrectly finding the common denominator or forgetting to multiply the numerator when adjusting fractions. Always check your work by substituting back into the original equation.