We need to solve (5/9)(x - 32) = 10 by isolating x. First, multiply both sides by 9/5 to clear the fraction: x - 32 = 10 × (9/5) = 90/5 = 18. Then add 32 to both sides to get x = 18 + 32 = 50. A common mistake is to multiply by 5/9 instead of its reciprocal 9/5, which would give x - 32 = 10 × (5/9) = 50/9 ≈ 5.56, leading to x ≈ 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 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 = (d - 12)/v. A common error is to divide by v before subtracting 12, which would give d/v = t + 12/v, leading to t = d/v - 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 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 ÷ 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 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.

To solve $T = 1.8C + 32$ for $C$, we need to isolate $C$. First, subtract 32 from both sides: $T - 32 = 1.8C$. Then divide both sides by 1.8: $C = \frac{T - 32}{1.8}$. The common error is trying to divide before subtracting 32, or only dividing one term by 1.8. When solving literal equations, follow the same order of operations as with numerical equations: undo addition/subtraction before undoing multiplication/division.

The equation $120 + 15x = 200 - 5x$ represents when two account balances are equal. Add $5x$ to both sides: $120 + 20x = 200$. Subtract 120 from both sides: $20x = 80$. Divide by 20: $x = 4$. The common mistake is subtracting $15x$ instead of adding $5x$, which would give a negative coefficient. When collecting variable terms, move them to create a positive coefficient when possible.

The equation $x + 12 = 18 + 0.5x$ represents when two phone plans cost the same. To solve, subtract $0.5x$ from both sides: $x - 0.5x + 12 = 18$, which gives us $0.5x + 12 = 18$. Subtract 12 from both sides: $0.5x = 6$. Divide by 0.5 (or multiply by 2): $x = 12$. The common error is incorrectly combining $x - 0.5x$ as $-0.5x$ instead of $0.5x$. Remember that $x = 1x$, so $1x - 0.5x = 0.5x$.

The equation $2(x+7.5)=41.5$ models the store's pricing where $x$ is the wholesale price. To solve, first divide both sides by 2: $x + 7.5 = 20.75$. Then subtract 7.5 from both sides: $x = 20.75 - 7.5 = 13.25$. The key error students make is trying to distribute the 2 instead of dividing by it first, or making arithmetic errors when subtracting decimals. When an entire expression in parentheses is multiplied by a constant, dividing by that constant first often simplifies the problem.