Multiplying each term by 12: $$\frac{3x-2}{4} \cdot 12 = 3(3x-2)$$, $$\frac{x+6}{3} \cdot 12 = 4(x+6)$$, and $$\frac{x-1}{2} \cdot 12 = 6(x-1)$$. The equation becomes $$3(3x-2) = 4(x+6) - 6(x-1)$$. Choice A shows the equation after distributing. Choice C has wrong sign after distributing $$-6(x-1)$$. Choice D changes subtraction to addition.

When you encounter equations with decimals, multiplying by a power of 10 is a smart strategy to eliminate decimal points and work with whole numbers. The key is applying this multiplication correctly to every term in the equation.

Starting with $$0.2x + 0.15(x-20) = 0.25(x+10)$$, when you multiply every term by 100, you need to convert each decimal coefficient. Since $$0.2 \times 100 = 20$$, $$0.15 \times 100 = 15$$, and $$0.25 \times 100 = 25$$, the equation becomes $$20x + 15(x-20) = 25(x+10)$$. Notice that the expressions in parentheses remain unchanged because you're only multiplying the coefficients outside.

Choice A incorrectly converts $$0.15$$ to $$1.5$$ and $$0.25$$ to $$2.5$$, suggesting multiplication by only 10 instead of 100. Choice B makes the opposite error—it correctly multiplies $$0.15$$ and $$0.25$$ by 100 but incorrectly converts $$0.2$$ to $$2$$, as if multiplying by only 10. Choice D contains a fundamental misunderstanding, incorrectly multiplying the numbers inside the parentheses (turning $$-20$$ into $$-2000$$ and $$+10$$ into $$+1000$$) while also using the wrong multiplier for the coefficients.

Study tip: When multiplying an equation by a constant, apply it only to the coefficients (the numbers in front of variables and parentheses), never to the terms inside parentheses. Always use the same multiplier consistently throughout the entire equation.

The LCD of 3, 6, and 4 is 12. Multiplying each term: $$\frac{2x-1}{3} \cdot 12 = 4(2x-1)$$, $$\frac{x+4}{6} \cdot 12 = 2(x+4)$$, and $$\frac{5x-2}{4} \cdot 12 = 3(5x-2)$$. Choice B uses 6 as LCD incorrectly. Choice C uses wrong multipliers. Choice D multiplies by wrong factors.

When you encounter equations with decimals, converting to fractions can make the algebra cleaner. The key is to eliminate all decimals by finding an appropriate common denominator.

To convert the original equation $$0.3(x-2) - 0.5(2x+1) = 1.2$$ to fractions, first recognize that these decimals have denominators of 10 when written as fractions: $$0.3 = \frac{3}{10}$$, $$0.5 = \frac{5}{10}$$, and $$1.2 = \frac{12}{10}$$.

Substituting these fractions gives us: $$\frac{3}{10}(x-2) - \frac{5}{10}(2x+1) = \frac{12}{10}$$

To combine the left side over a common denominator of 10: $$\frac{3(x-2) - 5(2x+1)}{10} = \frac{12}{10}$$

This matches answer choice D exactly.

Let's examine why the other choices are wrong. Choice A incorrectly uses addition instead of subtraction between the terms ($$3(x-2) + 5(2x+1)$$ instead of $$3(x-2) - 5(2x+1)$$). Choice B uses 100 as the denominator on the left and 120 as the numerator on the right, which would come from incorrectly treating the decimals as hundredths. Choice C multiplies the first term by 30 and the second by 50, which represents an error in converting $$0.3$$ and $$0.5$$ to equivalent fractions.

Study tip: When converting decimals to fractions, identify the place value of each decimal first. One decimal place means tenths, so use 10 as your common denominator and convert each coefficient accordingly.

Distributing: $$2.5(x + 4) = 2.5x + 10$$. The equation becomes $$2.5x + 10 - 1.8x = 3.2$$. Combining like terms: $$(2.5 - 1.8)x + 10 = 0.7x + 10$$. Therefore $$k = 10$$. Choice A results from calculation error in distribution. Choice C comes from forgetting to distribute to the constant. Choice D comes from doubling the distributed constant.

This problem tests solving linear equations with fraction coefficients using distributive property and combining like terms. Process: distribute a(bx+c)=abx+ac, collect like terms using common denominators, isolate variable by moving x terms to one side and constants to the other, then solve by dividing both sides. For this equation: distribute 3/4(x-8) = 3x/4 - 6, so the equation becomes 3x/4 - 6 + x/2 = 10. To combine x terms, find common denominator: 3x/4 + x/2 = 3x/4 + 2x/4 = 5x/4. The equation is now 5x/4 - 6 = 10, so 5x/4 = 16, giving x = 64/5. Common errors include distributing incorrectly (forgetting to multiply both terms) or adding fractions without finding common denominators.

This question tests solving linear equations with fraction/decimal coefficients using distributive property and combining like terms. The process involves distributing a(bx+c)=abx+ac (e.g., 2/3(x-6)=2x/3-4), collecting like terms (2x+3x=5x, 1/2x+1/4x=3/4x using common denominator), isolating the variable (move x terms one side, constants other), and solving (divide both sides); clearing fractions by multiplying by LCD simplifies arithmetic (equation with 1/2 and 3/4: multiply by 4 converts to integers). For the equation 0.5(x+6) + 1.25 = 0.8x - 0.25, distribute to get 0.5x + 3 + 1.25 = 0.8x - 0.25, combine constants to 0.5x + 4.25 = 0.8x - 0.25, subtract 0.5x from both sides to get 4.25 = 0.3x - 0.25, add 0.25 to both sides to get 4.5 = 0.3x, and divide by 0.3 to find x=15. The correct process yields x=15, which matches choice C. A common error is mishandling decimals, such as subtracting 0.5 from 0.8 incorrectly as 0.2 instead of 0.3, or forgetting to add the constants properly. Steps: (1) clear fractions if desired (multiply by LCD), (2) distribute (remove parentheses), (3) collect like terms (combine x's, combine constants), (4) isolate x (add/subtract to get x terms one side, constants other), (5) divide (coefficient of x), (6) simplify (reduce fraction if needed), (7) check (substitute back). Common errors: distributing only first term, adding fractions without common denominator (1/2+1/3≠2/5), sign errors moving terms, dividing wrong (x/4=2 → x=8 not 0.5).

This problem tests solving linear equations with decimals in a real-world context about mixing solutions. Process: distribute $0.25(4x-8) = x - 2$, combine constants, isolate variable, and solve. Starting with $0.25(4x-8) + 1.5 = 5.5$, distribute to get $x - 2 + 1.5 = 5.5$. Simplify left side: $x - 0.5 = 5.5$. Add 0.5 to both sides: $x = 6$. Common errors include incorrect distribution ($0.25 \times 4x = x$, not $0.25x$) or decimal arithmetic mistakes. Steps: (1) distribute carefully with decimals, (2) combine constants, (3) isolate x, (4) check by substituting back: $0.25(4\cdot6-8) + 1.5 = 0.25(16) + 1.5 = 4 + 1.5 = 5.5$ ✓.

This problem tests solving linear equations with fraction/decimal coefficients using distributive property and combining like terms. Process: distribute $a(bx+c) = abx + ac$ ($\frac{2}{3}(x-6) = \frac{2x}{3} - 4$), collect like terms ($2x + 3x = 5x$, $\frac{1}{2}x + \frac{1}{4}x = \frac{3}{4}x$ using common denominator), isolate variable (move x terms one side, constants other), solve (divide both sides). Clearing fractions by multiplying by LCD simplifies arithmetic (equation with $\frac{1}{2}$ and $\frac{3}{4}$: multiply by 4 converts to integers). For this specific equation, distribute $-\frac{1}{3}$ to (x-9) to get $- \frac{1}{3}x + 3$, then add $\frac{5}{6}x$ yielding $\frac{5}{6}x - \frac{1}{3}x + 3 = 12$; combine to $\frac{1}{2}x + 3 = 12$, subtract 3 to get $\frac{1}{2}x = 9$, then multiply by 2 to find x = 18. The correct process yields the answer $x = 18$. A common error is distributing with the wrong sign, like subtracting instead of adding 3, leading to wrong values like $x = 9$. Steps: (1) clear fractions if desired (multiply by LCD), (2) distribute (remove parentheses), (3) collect like terms (combine x's, combine constants), (4) isolate x (add/subtract to get x terms one side, constants other), (5) divide (coefficient of x), (6) simplify (reduce fraction if needed), (7) check (substitute back). Common errors: distributing only first term, adding fractions without common denominator ($\frac{1}{2} + \frac{1}{3} \neq \frac{2}{5}$), sign errors moving terms, dividing wrong ($ \frac{x}{4} = 2 \rightarrow x=8$ not 0.5).

This problem tests solving linear equations with fraction/decimal coefficients using distributive property and combining like terms. Process: distribute a(bx+c)=abx+ac (2/3(x-6)=2x/3-4), collect like terms (2x+3x=5x, 1/2x+1/4x=3/4x using common denominator), isolate variable (move x terms one side, constants other), solve (divide both sides). Clearing fractions by multiplying by LCD simplifies arithmetic (equation with 1/2 and 3/4: multiply by 4 converts to integers). For this specific equation, distribute 2.5 to (x-1.2) to get 2.5x - 3, then subtract 0.5x yielding 2.5x - 0.5x - 3 = 6.4; combine to 2x - 3 = 6.4, add 3 to get 2x = 9.4, then divide by 2 to find x = 4.7 or 47/10. The correct process yields the answer x = 47/10. A common error is miscalculating the distribution like 2.5*1.2 as 2.5 instead of 3, leading to wrong values like 37/10. Steps: (1) clear fractions if desired (multiply by LCD), (2) distribute (remove parentheses), (3) collect like terms (combine x's, combine constants), (4) isolate x (add/subtract to get x terms one side, constants other), (5) divide (coefficient of x), (6) simplify (reduce fraction if needed), (7) check (substitute back). Common errors: distributing only first term, adding fractions without common denominator (1/2+1/3≠2/5), sign errors moving terms, dividing wrong (x/4=2 → x=8 not 0.5).