Multi-Step Equations - SHSAT Math

1. Solve for $x$: $\dfrac{7x}{3}+5=20.$
   Answer: $\dfrac{45}{7}$ When you encounter a linear equation with fractions, your goal is to isolate the variable by using inverse operations in the correct order. Starting with $\frac{7x}{3} + 5 = 20$, first subtract 5 from both sides to eliminate the constant term: $\frac{7x}{3} = 15$. Next, multiply both sides by 3 to clear the fraction: $7x = 45$. Finally, divide both sides by 7 to solve for x: $x = \frac{45}{7}$. Let's examine why each answer choice is right or wrong. Choice A, $\frac{45}{7}$, is correct as we just demonstrated. Choice B, 6, likely comes from incorrectly treating the equation as $7x + 5 = 20$ (ignoring the denominator 3), which would give $x = \frac{15}{7} \times \frac{7}{3} = 5$, but that's still not 6—this suggests multiple computational errors. Choice C, $\frac{15}{7}$, results from forgetting to divide by 7 in the final step; students who get this stopped at $7x = 15$ instead of $7x = 45$. Choice D, $\frac{20}{3}$, appears to come from mishandling the fraction entirely, perhaps trying to cross-multiply incorrectly. Always verify your solution by substituting back into the original equation. With $x = \frac{45}{7}$: $\frac{7 \cdot \frac{45}{7}}{3} + 5 = \frac{45}{3} + 5 = 15 + 5 = 20$ ✓. On the SHSAT, fraction problems often include answer choices that represent common algebraic mistakes—double-check your arithmetic at each step.
2. Solve: $7(x-2)-3(x+4)+9=0.$
   Answer: $\dfrac{17}{4}$ When you encounter a linear equation with parentheses and multiple terms like this one, your goal is to systematically simplify by distributing, combining like terms, and isolating the variable. Start by distributing each coefficient to the terms inside the parentheses: $7(x-2) = 7x - 14$ and $-3(x+4) = -3x - 12$. This gives you: $7x - 14 - 3x - 12 + 9 = 0$. Next, combine like terms. The $x$ terms: $7x - 3x = 4x$. The constants: $-14 - 12 + 9 = -17$. Your equation becomes: $4x - 17 = 0$. Finally, solve for $x$ by adding 17 to both sides: $4x = 17$, so $x = \frac{17}{4}$. Let's examine why the other answers are wrong. Choice B ($\frac{19}{4}$) likely results from an error in combining constants—perhaps calculating $-14 - 12 + 9 = -15$ instead of $-17$. Choice C ($4$) might come from forgetting to distribute properly or from confusing the coefficient of $x$ with the final answer. Choice D ($\frac{15}{4}$) could result from sign errors during distribution, such as treating $-3(x+4)$ as $-3x + 12$ instead of $-3x - 12$. The correct answer is A: $\frac{17}{4}$. Study tip: Always work step-by-step with distribution problems: distribute first, then combine like terms, then isolate the variable. Double-check your arithmetic at each step, especially with negative signs—they're the most common source of errors in these problems.
3. Solve: $10-4\Bigl(1-\dfrac{x}{5}\Bigr)=6.$
   Answer: 0 When you encounter a linear equation with fractions and parentheses like this one, your goal is to systematically isolate the variable by working through the order of operations in reverse. Start by simplifying the left side. First, distribute the $-4$ through the parentheses: $10 - 4(1 - \frac{x}{5}) = 10 - 4 + \frac{4x}{5} = 6 + \frac{4x}{5}$. Now your equation becomes $6 + \frac{4x}{5} = 6$. Subtract 6 from both sides: $\frac{4x}{5} = 0$. To solve for $x$, multiply both sides by $\frac{5}{4}$: $x = 0 \cdot \frac{5}{4} = 0$. You can verify this by substituting back: $10 - 4(1 - \frac{0}{5}) = 10 - 4(1) = 6$ ✓ Choice A (0) is correct. Choice B (5) likely comes from incorrectly thinking that $\frac{4x}{5} = 0$ means $x = 5$, possibly from confusing the denominator with the solution. Choice C (2) might result from arithmetic errors when distributing or combining terms. Choice D (4) could come from mistakenly thinking the coefficient 4 in $\frac{4x}{5}$ becomes the answer, or from errors in the distribution step. Remember that when you have $\frac{ax}{b} = 0$ where $a \neq 0$ and $b \neq 0$, the solution is always $x = 0$. Any non-zero number multiplied by zero equals zero, regardless of what fraction it's part of. This pattern appears frequently on the SHSAT, so recognizing it quickly will save you time.
4. Solve: $-0.5(6x-8)+4=7.$
   Answer: $\dfrac13$ This is a linear equation with decimals and parentheses that requires careful order of operations and algebraic manipulation. When you encounter equations like this, work systematically through distribution, combining like terms, and isolating the variable. Start by distributing the $-0.5$ to both terms inside the parentheses: $-0.5(6x) + (-0.5)(-8) + 4 = 7$, which gives you $-3x + 4 + 4 = 7$. Combining like terms on the left side: $-3x + 8 = 7$. Subtract 8 from both sides: $-3x = -1$. Finally, divide both sides by $-3$: $x = \frac{-1}{-3} = \frac{1}{3}$. This matches choice A. Let's examine why the other answers are incorrect. Choice B ($\frac{2}{3}$) likely results from an error in the final division step, perhaps incorrectly calculating $\frac{-1}{-3}$ or making an arithmetic mistake earlier. Choice C ($-\frac{1}{3}$) comes from forgetting that dividing two negative numbers gives a positive result—this is what you'd get if you calculated $\frac{-1}{-3} = -\frac{1}{3}$. Choice D ($-\frac{2}{3}$) combines both errors: incorrect signs and wrong numerator. You can verify: substituting $x = \frac{1}{3}$ back into the original equation gives $-0.5(6 \cdot \frac{1}{3} - 8) + 4 = -0.5(2-8) + 4 = -0.5(-6) + 4 = 3 + 4 = 7$ ✓ Strategy tip: Always distribute carefully with decimals, and remember that dividing two negative numbers yields a positive result. Check your answer by substituting back into the original equation.
5. Solve: $\dfrac25(15-10x)+7=-1.$
   Answer: 3.5 When solving linear equations with fractions and parentheses, work systematically through the order of operations while isolating the variable step by step. Start by distributing the fraction $\frac{2}{5}$ to both terms inside the parentheses: $\frac{2}{5}(15-10x) = \frac{2}{5} \cdot 15 - \frac{2}{5} \cdot 10x = 6 - 4x$ The equation becomes: $6 - 4x + 7 = -1$ Combine like terms on the left side: $13 - 4x = -1$ Subtract 13 from both sides: $-4x = -14$ Divide both sides by -4: $x = \frac{14}{4} = 3.5$ Choice A (3.5) is correct. Choice B (2.5) likely results from an error when combining constants—perhaps calculating $6 + 7 = 11$ instead of 13, leading to $-4x = -12$ and $x = 3$. Wait, that would give 3, not 2.5. Choice B probably comes from a sign error or incorrect distribution. Choice D (3) occurs when you make an arithmetic mistake with the constants, getting $-4x = -12$. Choice C (4) might result from incorrectly distributing the fraction or making a sign error during isolation. You can verify: $\frac{2}{5}(15-10(3.5))+7 = \frac{2}{5}(15-35)+7 = \frac{2}{5}(-20)+7 = -8+7 = -1$ ✓ Strategy tip: Always distribute fractions carefully to each term in parentheses, and verify your answer by substituting back into the original equation. This catches arithmetic errors that commonly appear in SHSAT problems.
6. Solve: $\dfrac23(9-6x)=4.$
   Answer: $\dfrac12$ This is a linear equation that requires you to isolate the variable through systematic algebraic steps. When you see fractions multiplied by expressions in parentheses, your goal is to clear the fraction first, then work with the resulting simpler equation. Start by multiplying both sides by $\frac{3}{2}$ (the reciprocal of $\frac{2}{3}$) to eliminate the fraction: $9 - 6x = 4 \cdot \frac{3}{2} = 6$. Now you have the simpler equation $9 - 6x = 6$. Subtract 9 from both sides: $-6x = -3$. Finally, divide both sides by -6: $x = \frac{-3}{-6} = \frac{1}{2}$. This confirms that A) $\frac{1}{2}$ is correct. Let's examine why the other answers are wrong. Choice B) $1$ likely comes from making an error when clearing the fraction or solving $-6x = -3$—perhaps dividing by -3 instead of -6. Choice C) $\frac{1}{3}$ might result from incorrectly handling the fraction multiplication or making a sign error during the solving process. Choice D) $\frac{2}{3}$ could come from confusing this value with the original coefficient in the problem, or from computational mistakes when working with the fractions. To verify your answer, always substitute back into the original equation: $\frac{2}{3}(9 - 6 \cdot \frac{1}{2}) = \frac{2}{3}(9 - 3) = \frac{2}{3}(6) = 4$ ✓ Strategy tip: When solving equations with fractions, eliminate the fraction first by multiplying by its reciprocal. This makes the algebra much cleaner and reduces calculation errors.
7. Solve for $x$: $-3(4x+5)+2=-10.$
   Answer: $-\dfrac14$ When you encounter a linear equation with parentheses and multiple terms, your goal is to systematically isolate the variable using inverse operations in the proper order. Start by distributing the $-3$ across the parentheses: $-3(4x+5) = -12x - 15$. This gives you $-12x - 15 + 2 = -10$. Combine like terms on the left: $-12x - 13 = -10$. Add 13 to both sides: $-12x = -10 + 13 = 3$. Finally, divide both sides by $-12$: $x = \frac{3}{-12} = -\frac{1}{4}$. Let's verify: $-3(4(-\frac{1}{4})+5)+2 = -3(-1+5)+2 = -3(4)+2 = -12+2 = -10$ ✓ Looking at the wrong answers: Choice B gives $\frac{1}{4}$, which you'd get if you forgot the negative sign when dividing by $-12$. Choice C gives $-\frac{3}{4}$, which results from incorrectly distributing or combining like terms—perhaps getting $-4x = 3$ instead of $-12x = 3$. Choice D gives $\frac{3}{4}$, combining the sign error from choice B with the coefficient error from choice C. The correct answer is A: $-\frac{1}{4}$. Strategy tip: Always distribute carefully and track your negative signs. After solving, substitute your answer back into the original equation to catch arithmetic mistakes—this verification step is especially valuable on timed exams when you might rush through calculations.
8. Solve: $6-2\bigl[4-3(x-2)\bigr]=-18.$
   Answer: $-\dfrac23$ When you encounter a multi-step equation with nested brackets like this, your success depends on working systematically from the inside out, following the order of operations. Start with the innermost parentheses: $3(x-2) = 3x - 6$. Now substitute this back: $6 - 2[4 - (3x - 6)] = -18$. Simplify inside the brackets: $4 - (3x - 6) = 4 - 3x + 6 = 10 - 3x$. The equation becomes: $6 - 2(10 - 3x) = -18$. Distribute the $-2$: $6 - 20 + 6x = -18$, which simplifies to $-14 + 6x = -18$. Add 14 to both sides: $6x = -4$. Therefore: $x = -\frac{4}{6} = -\frac{2}{3}$. Looking at the wrong answers: Choice B ($-1$) likely comes from making an error when distributing the negative sign or combining like terms. Choice C ($\frac{2}{3}$) is the positive version of the correct answer—a common mistake when students lose track of negative signs during the multi-step process. Choice D ($1$) might result from computational errors in the final division step or earlier sign errors. The correct answer is A: $-\frac{2}{3}$. Strategy tip: With nested grouping symbols, always work from the innermost outward and double-check your sign changes at each step. Many SHSAT algebra errors happen when students rush through the distribution of negative signs—slow down and verify each transformation.
9. Sarah needs to solve $7 - 2(3x - 4) = 5x + 3$. After correctly distributing the $-2$, she gets $7 - 6x + 8 = 5x + 3$. What value of $x$ makes this equation true?
   Answer: $x = \frac{12}{11}$ Starting from Sarah's work: $7 - 6x + 8 = 5x + 3$. Combine like terms on the left: $15 - 6x = 5x + 3$. Add $6x$ to both sides: $15 = 11x + 3$. Subtract 3 from both sides: $12 = 11x$. Divide by 11: $x = \frac{12}{11}$. Choice B results from the error $8 = 11x$. Choice C comes from incorrectly getting $11 = 12x$. Choice D results from the mistake $15 = 11x$ without subtracting 3.
10. Solve: $\dfrac35(x+10)-2=4.$
    Answer: 0 This is a linear equation that requires you to isolate the variable $x$ using inverse operations in the proper order. Start by adding 2 to both sides to eliminate the constant term: $\frac{3}{5}(x+10) = 6$. Next, multiply both sides by $\frac{5}{3}$ (the reciprocal of $\frac{3}{5}$) to clear the fraction: $x + 10 = 6 \times \frac{5}{3} = 10$. Finally, subtract 10 from both sides: $x = 0$. Let's verify: $\frac{3}{5}(0+10) - 2 = \frac{3}{5}(10) - 2 = 6 - 2 = 4$ ✓ Choice A (0) is correct. Choice B (2) would give you $\frac{3}{5}(12) - 2 = 7.2 - 2 = 5.2$, not 4. This might result from incorrectly thinking $x + 10 = 12$ when you have $x + 10 = 10$. Choice C (5) would yield $\frac{3}{5}(15) - 2 = 9 - 2 = 7$, which could happen if you forgot to multiply by the reciprocal $\frac{5}{3}$ and instead just multiplied by 5. Choice D (-2) gives $\frac{3}{5}(8) - 2 = 4.8 - 2 = 2.8$, which might occur from calculation errors when working with the fraction. When solving equations with fractions, always use the reciprocal to eliminate the fraction completely rather than trying to work around it. This prevents calculation errors and makes the algebra much cleaner.
11. Solve: $11-\dfrac{x}{3}=8.$
    Answer: 9 This is a one-step linear equation that requires isolating the variable $x$. When you see an equation with a fraction involving the variable, your goal is to eliminate the fraction by using inverse operations. Starting with $11-\dfrac{x}{3}=8$, first isolate the fraction term by subtracting 11 from both sides: $-\dfrac{x}{3} = 8-11 = -7$. Now you have $-\dfrac{x}{3} = -7$. To solve for $x$, multiply both sides by $-3$: $x = (-7) \times (-3) = 21$. Wait—21 isn't among the choices, so let me check this calculation. Actually, let's verify by substituting our potential answers back into the original equation. For choice A) $x = 9$: $11-\dfrac{9}{3} = 11-3 = 8$ ✓ For choice B) $x = 6$: $11-\dfrac{6}{3} = 11-2 = 9 \neq 8$ For choice C) $x = 3$: $11-\dfrac{3}{3} = 11-1 = 10 \neq 8$ For choice D) $x = 12$: $11-\dfrac{12}{3} = 11-4 = 7 \neq 8$ The answer is A) 9. Going back to my algebra: from $-\dfrac{x}{3} = -7$, I should multiply both sides by $-3$ to get $x = 21$... but that's not matching. Let me recalculate: $11-8=3$, so $\dfrac{x}{3}=3$, which means $x=9$. Choices B, C, and D all produce different values when substituted back into the original equation, showing they're incorrect solutions. Strategy tip: Always verify your answer by substituting back into the original equation—this catches arithmetic errors and confirms your solution works.
12. Solve: $8-(3x-4)=2.$
    Answer: $\dfrac{10}{3}$ This is a linear equation with parentheses that requires careful attention to the order of operations and distribution rules. When you see parentheses in an equation, you need to decide whether to distribute first or use other algebraic techniques. Let's solve $8-(3x-4)=2$ step by step. First, distribute the negative sign to both terms inside the parentheses: $8-3x+4=2$. Combine like terms on the left side: $12-3x=2$. Subtract 12 from both sides: $-3x=-10$. Finally, divide both sides by $-3$: $x=\frac{10}{3}$. Let's verify: $8-(3 \cdot \frac{10}{3}-4) = 8-(10-4) = 8-6 = 2$ ✓ Looking at the wrong answers: Choice B ($\frac{5}{2}$) likely comes from making an error when combining like terms or forgetting to distribute the negative sign properly. Choice C ($3$) might result from incorrectly handling the fraction arithmetic or making a sign error during the solving process. Choice D ($\frac{7}{3}$) could come from mistakes in the distribution step or errors when isolating the variable. The correct answer is A: $\frac{10}{3}$. Strategy tip: When distributing a negative sign across parentheses, be extra careful with signs. A common mistake is forgetting that $-(3x-4) = -3x+4$, not $-3x-4$. Always double-check by substituting your answer back into the original equation—this catches most algebraic errors quickly.
13. Marcus is solving the equation $3(2x - 5) + 2x = 2x + 7$. After distributing and combining like terms on the left side, he gets $8x - 15 = 2x + 7$. What value of $x$ satisfies the original equation?
    Answer: $x = \frac{11}{3}$ Marcus's first step is right for this equation. Distribute: $3(2x - 5) = 6x - 15$. Then add $2x$: $6x - 15 + 2x = 8x - 15$. So the equation is $8x - 15 = 2x + 7$. Subtract $2x$ from both sides: $6x - 15 = 7$. Add 15 to both sides: $6x = 22$. Divide by 6: $x = \frac{22}{6} = \frac{11}{3}$. Check: $3\left(\frac{22}{3} - 5\right) + \frac{22}{3} = 7 + \frac{22}{3} = \frac{43}{3}$, and $\frac{22}{3} + 7 = \frac{43}{3}$. Adding $2x$ to the left instead of subtracting it gives $10x - 15 = 7$, or $x = \frac{11}{5}$. Subtracting 15 from 7 instead of adding gives $6x = -8$, a negative answer. Trying $x = 4$ gives $25 = 15$, so it does not work. The value of $x$ is $\frac{11}{3}$.
14. Solve: $4x-7(2-x)=45.$
    Answer: $\dfrac{59}{11}$ This is a linear equation that requires you to distribute, combine like terms, and isolate the variable. When you see an equation with parentheses, your first step should always be to distribute to eliminate them. Start by distributing the $-7$ to both terms inside the parentheses: $4x - 7(2) - 7(-x) = 45$, which gives you $4x - 14 + 7x = 45$. Now combine like terms on the left side: $11x - 14 = 45$. Add 14 to both sides: $11x = 59$. Finally, divide both sides by 11: $x = \frac{59}{11}$. Looking at the wrong answers, choice B ($\frac{45}{11}$) likely comes from forgetting to add 14 to the right side after getting $11x - 14 = 45$, incorrectly concluding that $11x = 45$. Choice C ($\frac{41}{11}$) probably results from a sign error when distributing—if you mistakenly got $4x - 14 - 7x = 45$, you'd have $-3x - 14 = 45$, leading to $-3x = 59$ and $x = -\frac{59}{3}$, though this doesn't match exactly. Choice D ($\frac{63}{11}$) might come from an arithmetic error when combining like terms or solving. Remember: distribution problems on the SHSAT often include negative signs that can trip you up. Always be extra careful with signs when distributing, and double-check your work by substituting your answer back into the original equation to verify it's correct.
15. Solve: $\dfrac{5x-2}{4}+3=8.$
    Answer: $\dfrac{22}{5}$ When you encounter a linear equation with fractions, your goal is to isolate the variable by systematically undoing the operations in reverse order. Starting with $\dfrac{5x-2}{4}+3=8$, first eliminate the constant term by subtracting 3 from both sides: $\dfrac{5x-2}{4}=5$. Next, clear the fraction by multiplying both sides by 4: $5x-2=20$. Add 2 to both sides: $5x=22$. Finally, divide by 5: $x=\dfrac{22}{5}$. Let's examine why each answer choice might appear. Choice A, $\dfrac{22}{5}$, is correct as shown above. Choice B, $\dfrac{11}{2}$, likely results from making an error when clearing the fraction—perhaps multiplying only the numerator by 4 instead of the entire left side, leading to $5x-2=20$ becoming $20x-2=20$, then $x=\dfrac{22}{20}=\dfrac{11}{10}$, or from other algebraic mistakes. Choice C, $\dfrac{24}{5}$, might come from incorrectly handling the subtraction step, perhaps adding 2 instead of subtracting when isolating $5x$. Choice D, $\dfrac{23}{5}$, could result from arithmetic errors in the final steps or confusion about order of operations. The key strategy here is working systematically: handle addition/subtraction before multiplication/division, and always perform the same operation to both sides. When dealing with fractions in equations, multiply both sides by the denominator to clear them early—this prevents mistakes and simplifies your work.
16. Solve: $-2\bigl[3-5(x-2)\bigr]=28.$
    Answer: $\dfrac{27}{5}$ This is a multi-step linear equation that requires careful attention to the order of operations and algebraic manipulation. When you see nested brackets with multiple operations, work systematically from the inside out. Start by expanding the expression inside the brackets. First, distribute the $-5$ to both terms in $(x-2)$: $-5(x-2) = -5x + 10$. Now the expression becomes $3 - (-5x + 10) = 3 + 5x - 10 = 5x - 7$. So your equation is now $-2(5x - 7) = 28$. Distribute the $-2$: $-10x + 14 = 28$. Subtract $14$ from both sides: $-10x = 14$. Finally, divide by $-10$: $x = -\frac{14}{10} = -\frac{7}{5}$. Wait—this doesn't match any answer choice! Let me recalculate more carefully. From $3 - 5(x-2)$: the $-5$ distributes to get $3 - 5x + 10 = 13 - 5x$. Now $-2(13 - 5x) = 28$ becomes $-26 + 10x = 28$. Adding $26$ to both sides: $10x = 54$, so $x = \frac{54}{10} = \frac{27}{5}$. Choice A ($\frac{27}{5}$) is correct. Choice B ($\frac{14}{5}$) likely comes from forgetting to distribute the $-2$ to the constant term. Choice C ($\frac{11}{2}$) might result from sign errors when distributing. Choice D ($5$) could come from calculation mistakes or improper fraction reduction. Remember: when solving equations with nested operations, distribute step-by-step and track your signs carefully. Double-check by substituting your answer back into the original equation.
17. Solve: $5-[2x-(3-x)]= -2.$
    Answer: $\dfrac{10}{3}$ This problem tests your ability to work with nested brackets and negative signs—two areas where students commonly make errors on the SHSAT. Start by carefully distributing the negative sign in front of the brackets. The expression $-[2x-(3-x)]$ becomes $-(2x-(3-x))$, which equals $-2x+(3-x)$ or $-2x+3-x = -3x+3$. So your equation becomes: $5+(-3x+3) = -2$, which simplifies to $8-3x = -2$. Solving: $8-3x = -2$, so $-3x = -10$, giving us $x = \frac{10}{3}$. This matches choice A. Let's examine why the other answers are incorrect. Choice B ($x = 3$) likely comes from incorrectly handling the nested parentheses—perhaps writing $-[2x-(3-x)]$ as $-2x-3+x$ instead of $-2x+3-x$. Choice C ($x = \frac{5}{2}$) could result from sign errors when isolating $x$, such as dividing $-10$ by $-4$ instead of $-3$. Choice D ($x = 4$) might come from completely mishandling the bracket distribution, leading to an equation like $2-3x = -2$. When you encounter problems with nested brackets and negative signs, work methodically: first distribute any negative signs outside brackets, then combine like terms inside parentheses, and finally solve step-by-step. Always double-check your work by substituting your answer back into the original equation.
18. Solve: $\dfrac{4x-1}{3}-\dfrac{2}{9}=5.$
    Answer: $\dfrac{25}{6}$ When you encounter an equation with fractions, your goal is to isolate the variable by systematically eliminating the fractions and simplifying. This question tests your ability to work with linear equations containing fractions. Start by getting all terms with fractions on one side. Add $\frac{2}{9}$ to both sides: $\frac{4x-1}{3} = 5 + \frac{2}{9}$ To add $5 + \frac{2}{9}$, convert 5 to ninths: $5 = \frac{45}{9}$. So: $\frac{4x-1}{3} = \frac{45}{9} + \frac{2}{9} = \frac{47}{9}$ Now multiply both sides by 3 to eliminate the remaining fraction: $4x - 1 = 3 \cdot \frac{47}{9} = \frac{141}{9} = \frac{47}{3}$ Add 1 to both sides: $4x = \frac{47}{3} + 1 = \frac{47}{3} + \frac{3}{3} = \frac{50}{3}$ Finally, divide by 4: $x = \frac{50}{3} \div 4 = \frac{50}{3} \cdot \frac{1}{4} = \frac{50}{12} = \frac{25}{6}$ This confirms answer A) $\frac{25}{6}$. Answer B) $\frac{50}{9}$ likely results from incorrectly adding fractions in the early steps. Answer C) $\frac{23}{6}$ suggests an arithmetic error when combining $\frac{47}{3} + 1$. Answer D) $\frac{11}{2}$ might come from mishandling the fraction operations or making errors in the final division step. Strategy tip: Always work systematically with fraction equations: clear fractions step by step, convert whole numbers to fractions with common denominators, and double-check your arithmetic at each stage.
19. Solve: $0.75x+1.5(4-x)=6.$
    Answer: 0 When you encounter a linear equation with decimals and parentheses, your goal is to simplify systematically using the distributive property and combining like terms. Start by distributing the 1.5 to both terms inside the parentheses: $0.75x + 1.5(4) + 1.5(-x) = 6$, which becomes $0.75x + 6 - 1.5x = 6$. Next, combine the like terms on the left side. The x-terms are $0.75x - 1.5x = -0.75x$, so you have $-0.75x + 6 = 6$. Subtract 6 from both sides: $-0.75x = 0$. Dividing both sides by -0.75 gives you $x = 0$. Let's verify: substituting $x = 0$ into the original equation gives $0.75(0) + 1.5(4-0) = 0 + 1.5(4) = 6$. ✓ Looking at the wrong answers: Choice B ($x = 2$) would give $0.75(2) + 1.5(2) = 1.5 + 3 = 4.5 \neq 6$. Choice C ($x = 4$) would yield $0.75(4) + 1.5(0) = 3 \neq 6$. Choice D ($x = 6$) produces $0.75(6) + 1.5(-2) = 4.5 - 3 = 1.5 \neq 6$. These wrong answers likely come from calculation errors in distributing or combining like terms. Study tip: When working with decimal coefficients, consider converting to fractions (0.75 = 3/4, 1.5 = 3/2) if that feels more comfortable, or always double-check your arithmetic when combining decimal terms.
20. Solve: $9-2(3-4x)=1.$
    Answer: $-\dfrac14$ This is a linear equation that requires you to use the distributive property and inverse operations to isolate the variable. When you see parentheses with a coefficient in front, always distribute first before combining like terms. Start by distributing the $-2$ to both terms inside the parentheses: $9 - 2(3 - 4x) = 9 - 6 + 8x = 3 + 8x$. So your equation becomes $3 + 8x = 1$. Next, subtract 3 from both sides: $8x = -2$. Finally, divide both sides by 8: $x = -\frac{2}{8} = -\frac{1}{4}$. Choice A gives $-\frac{1}{4}$, which matches our solution. Choice B gives $\frac{1}{4}$, which you'd get if you made a sign error when distributing the negative or when solving $8x = -2$. Choice C gives $-\frac{1}{2}$, which results from forgetting to distribute the $-2$ to the $4x$ term, leaving you with $9 - 6 - 4x = 1$, so $3 - 4x = 1$ and $x = \frac{1}{2}$, then incorrectly applying a negative sign. Choice D gives $\frac{1}{2}$, which comes from the same distribution error but without the final sign mistake. The key strategy here is to work methodically: distribute first, combine like terms, then use inverse operations. Double-check your distribution step since that's where most errors occur—make sure you apply the outside coefficient to every term inside the parentheses, keeping track of signs carefully.

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