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Solve the following linear equation for ww, paying careful attention to sign distribution across grouped fractional terms: 5−(3w−2)=4−2(5−3w)35 - (3w - 2) = 4 - \frac{2(5 - 3w)}{3}.

Answer and steps

Final answer

w=1915w = \frac{19}{15}

Steps

01

Simplify the left-hand side

To solve the linear equation for ww, follow these systematic steps:

Distribute the negative sign through the parentheses on the left: 5−(3w−2)=5−3w+2=7−3w5 - (3w - 2) = 5 - 3w + 2 = 7 - 3w

The equation now reads: 7−3w=4−2(5−3w)37 - 3w = 4 - \frac{2(5 - 3w)}{3}

02

Clear the fraction

To eliminate the denominator 33, multiply every term on both sides of the equation by 33: 3(7−3w)=3(4−2(5−3w)3)3(7 - 3w) = 3\left(4 - \frac{2(5 - 3w)}{3}\right) 21−9w=12−2(5−3w)21 - 9w = 12 - 2(5 - 3w)

03

Expand the grouped terms on the right-hand side

Expand 2(5−3w)2(5 - 3w) to get 10−6w10 - 6w.

Common Pitfall: When subtracting a grouped expression, the negative sign applies to every term inside the group. 12−(10−6w)=12−10−(−6w)=2+6w12 - (10 - 6w) = 12 - 10 - (-6w) = 2 + 6w

Now the equation simplifies to: 21−9w=2+6w21 - 9w = 2 + 6w

04

Isolate the variable ww

Group all variable terms on one side and constant terms on the other:

  1. Subtract 6w6w from both sides: 21−15w=221 - 15w = 2
  2. Subtract 2121 from both sides: −15w=2−21-15w = 2 - 21 −15w=−19-15w = -19
  3. Divide both sides by −15-15: w=−19−15=1915w = \frac{-19}{-15} = \frac{19}{15}
Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student fails to distribute the negative sign to the second term inside the parentheses on the left-hand side, writing 5 - 3w - 2 instead of 5 - 3w + 2, resulting in w = 7/15.

Sample Student Work

Reviewed by ACEpath Teacher
Step-by-Step Review
Walk Through It with Your Teacher
Incorrect
  1. Step 1

    5 - 3w - 2 = 4 - 10−6w3\frac{10 - 6w}{3}   ⟹  \implies 3 - 3w = 4 - 10−6w3\frac{10 - 6w}{3}

    Correct

    5 - 3w + 2 = 4 - 10−6w3\frac{10 - 6w}{3}   ⟹  \implies 7 - 3w = 4 - 10−6w3\frac{10 - 6w}{3}

    When distributing the negative sign across −(3w−2)-(3w - 2), each term inside changes sign: −(3w)−(−2)=−3w+2-(3w) - (-2) = -3w + 2. Combining like terms gives 5+2−3w=7−3w5 + 2 - 3w = 7 - 3w, rather than 3−3w3 - 3w.

  2. Step 2

    9 - 9w = 12 - (10 - 6w)   ⟹  \implies 9 - 9w = 2 + 6w

    Correct

    21 - 9w = 12 - (10 - 6w)   ⟹  \implies 21 - 9w = 2 + 6w

    Your technique for clearing the fraction and distributing the negative sign on the right side was completely correct, but the left side was affected by the initial simplification error.

  3. Step 3

    -15w = -7

    Correct

    -15w = -19

    You correctly subtracted 6w6w and 99 from both sides, but the calculation followed from the earlier incorrect constant.

  4. Step 4

    w = 715\frac{7}{15}

    Correct

    w = 1915\frac{19}{15}

    You correctly divided by −15-15, arriving at 715\frac{7}{15} based on your previous values.

WHAT YOU DID WELL
Compliments from Your Teacher

You correctly multiplied both sides of the equation by 33 to clear the fractional denominator.

You properly grouped and distributed the subtraction on the right-hand side as 12−(10−6w)=2+6w12 - (10 - 6w) = 2 + 6w.

You accurately collected like terms and isolated the variable using inverse operations.

THE ERROR
Where Your ACEpath Teacher Spotted the Mistake
Sign distribution error

You forgot to distribute the negative sign to the second term inside −(3w−2)-(3w - 2), treating it as −2-2 instead of +2+2.

WHY IT HAPPENED
ACEpath Teacher's Diagnosis

Treating a minus sign in front of parentheses as applying only to the immediately adjacent leading term rather than distributing across all terms inside the grouped expression.

REASONING PATH
Your ACEpath teacher’s suggested approach
Distribute to every term

  • −(a−b)=−a−(−b)=−a+b-(a - b) = -a - (-b) = -a + b

  • Rewrite a leading minus sign as multiplying by −1-1, so −(3w−2)-(3w - 2) explicitly becomes (−1)(3w)+(−1)(−2)=−3w+2(-1)(3w) + (-1)(-2) = -3w + 2.

CORRECTED PATH
ACEpath Teacher's Demonstration

  1. 5 - (3w - 2) = 4 - 2(5−3w)3\frac{2(5 - 3w)}{3}

  2. 5 - 3w + 2 = 4 - 10−6w3\frac{10 - 6w}{3}

  3. 7 - 3w = 4 - 10−6w3\frac{10 - 6w}{3}

  4. 3(7 - 3w) = 12 - (10 - 6w)

  5. 21 - 9w = 2 + 6w

  6. -15w = -19

  7. w = 1915\frac{19}{15}

Final answer

w = 1915\frac{19}{15}

NEXT STEPS
Final Takeaways from Your Teacher

  • Practice rewriting expressions with a negative sign outside parentheses by inserting an explicit factor of −1-1.

  • Perform a quick mental sign check on each term inside parentheses before proceeding to solve the rest of the equation.

OverviewACEpath Teacher's Diagnosis

02 The student fails to distribute the negative sign to the second term in the numerator when clearing the fraction, writing -10 - 6w instead of -10 + 6w, resulting in w = 1/3.

Sample Student Work

Reviewed by ACEpath Teacher
Step-by-Step Review
Walk Through It with Your Teacher
Incorrect
  1. Step 1

    7 - 3w = 4 - 10−6w3\frac{10 - 6w}{3}

  2. Step 2

    21 - 9w = 12 - 10 - 6w

    Correct

    21 - 9w = 12 - (10 - 6w)   ⟹  \implies 21 - 9w = 12 - 10 + 6w

    When clearing the fraction by multiplying by 33, the minus sign in front of the fraction applies to the entire numerator: −10−6w3⋅3=−(10−6w)=−10+6w-\frac{10 - 6w}{3} \cdot 3 = -(10 - 6w) = -10 + 6w.

  3. Step 3

    21 - 9w = 2 - 6w   ⟹  \implies -3w = -19

    Correct

    21 - 9w = 2 + 6w   ⟹  \implies -15w = -19

    Your algebraic rearrangement was carried out correctly according to your previous line, but it inherited the sign error from the previous step.

  4. Step 4

    w = 193\frac{19}{3}

    Correct

    w = 1915\frac{19}{15}

    You solved for ww consistently from your isolated equation, but the final value is incorrect due to the propagated sign error.

WHAT YOU DID WELL
Compliments from Your Teacher

You correctly simplified 5−(3w−2)5 - (3w - 2) to 7−3w7 - 3w.

You correctly expanded 2(5−3w)2(5 - 3w) to 10−6w10 - 6w in the numerator.

You multiplied every term by the common denominator 33 to eliminate the fraction.

You maintained algebraic consistency when moving terms across the equals sign.

THE ERROR
Where Your ACEpath Teacher Spotted the Mistake
Sign distribution over fraction numerator

You treated the subtraction bar of the fraction as applying only to the first term 1010, forgetting that a fraction bar acts as grouping parentheses for the whole numerator.

WHY IT HAPPENED
ACEpath Teacher's Diagnosis

Overlooking that a fraction bar functions as an implicit grouping symbol; when clearing a fraction preceded by a minus sign, parentheses must be preserved around the numerator until the negative sign is fully distributed.

REASONING PATH
Your ACEpath teacher’s suggested approach
Fraction Bar as Grouping Symbol

  • −A−BC=−(A−B)C-\frac{A - B}{C} = -\frac{(A - B)}{C}

  • Always write explicit parentheses around the numerator whenever a fraction is preceded by a minus sign before clearing denominators.

Distribute the Negative Sign

  • −(a−b)=−a+b-(a - b) = -a + b

  • Change the sign of every term inside the numerator when removing the enclosing parentheses.

CORRECTED PATH
ACEpath Teacher's Demonstration

  1. Multiply both sides by 33: 3(7−3w)=3(4)−(10−6w)3(7 - 3w) = 3(4) - (10 - 6w)

  2. Distribute the negative sign over the grouped numerator: 21−9w=12−10+6w21 - 9w = 12 - 10 + 6w

  3. Combine constant terms on the right-hand side: 21−9w=2+6w21 - 9w = 2 + 6w

  4. Subtract 6w6w and subtract 2121 from both sides: −15w=−19-15w = -19

  5. Divide by −15-15: w=−19−15=1915w = \frac{-19}{-15} = \frac{19}{15}

Final answer

w = 1915\frac{19}{15}

NEXT STEPS
Final Takeaways from Your Teacher

  • Whenever a minus sign sits directly in front of a fraction, rewrite the numerator inside parentheses: −A+BC→−(A+B)C-\frac{A + B}{C} \rightarrow -\frac{(A + B)}{C}.

  • Practice distributing negative signs across multi-term numerators when clearing fractions.

  • Substitute your final answer back into the original equation to verify that both sides equate.

OverviewACEpath Teacher's Diagnosis

03 The student forgets to multiply the standalone constant 4 by 3 when clearing the fraction with denominator 3, resulting in w = 11/15.

Sample Student Work

Reviewed by ACEpath Teacher
Step-by-Step Review
Walk Through It with Your Teacher
Incorrect
  1. Step 1

    7 - 3w = 4 - (10−6w)3\frac{(10 - 6w)}{3}

  2. Step 2

    21 - 9w = 4 - (10 - 6w)

    Correct

    21 - 9w = 12 - (10 - 6w)

    When multiplying both sides of an equation by 33 to eliminate the fraction, every term on both sides must be multiplied by 33. The standalone term 44 must become 3×4=123 \times 4 = 12.

  3. Step 3

    21 - 9w = 4 - 10 + 6w   ⟹  \implies 21 - 9w = -6 + 6w

    Correct

    21 - 9w = 12 - 10 + 6w   ⟹  \implies 21 - 9w = 2 + 6w

    Your sign distribution and arithmetic follow consistently from the previous line, but the result is affected by the missing factor of 33 on the constant.

  4. Step 4

    -15w = -27   ⟹  \implies w = 2715\frac{27}{15} = 95\frac{9}{5}

    Correct

    -15w = -19   ⟹  \implies w = 1915\frac{19}{15}

    You solved for ww logically from your intermediate equation, but the final value reflects the earlier error.

WHAT YOU DID WELL
Compliments from Your Teacher

You distributed the negative sign across (3w−2)(3w - 2) correctly on the left side.

You handled the subtraction of the binomial −(10−6w)=−10+6w-(10 - 6w) = -10 + 6w with correct sign flips.

You isolated the variable and simplified the final fraction accurately.

THE ERROR
Where Your ACEpath Teacher Spotted the Mistake
Incomplete multiplication across terms when clearing fractions

You cleared the denominator of 33 from the fraction and multiplied the left side by 33, but left the term 44 unchanged instead of multiplying it to get 1212.

WHY IT HAPPENED
ACEpath Teacher's Diagnosis

Treating fraction clearing as canceling the denominator locally rather than applying the multiplication property of equality across every separated term on both sides of the equation.

REASONING PATH
Your ACEpath teacher’s suggested approach
Box Every Term

  • An equation consists of distinct terms separated by ++ or −- signs outside grouping symbols.

  • Before clearing a fraction, place parentheses or boxes around every separate term on both sides to ensure none are skipped.

Distribute the Common Denominator

  • Multiplying an equation by a number means multiplying each individual term by that number: c(a−b)=ca−cbc(a - b) = ca - cb.

  • Write the factor explicitly next to each term: 3(7)−3(3w)=3(4)−3(…3)3(7) - 3(3w) = 3(4) - 3\left(\frac{\dots}{3}\right).

CORRECTED PATH
ACEpath Teacher's Demonstration

  1. Multiply every term on both sides by 33: 3(7−3w)=3(4)−3(10−6w3)3(7 - 3w) = 3(4) - 3\left(\frac{10 - 6w}{3}\right)

  2. Simplify each term: 21−9w=12−(10−6w)21 - 9w = 12 - (10 - 6w)

  3. Distribute the negative sign: 21−9w=12−10+6w21 - 9w = 12 - 10 + 6w

  4. Combine like terms on the right-hand side: 21−9w=2+6w21 - 9w = 2 + 6w

  5. Subtract 6w6w and subtract 2121 from both sides: −15w=−19-15w = -19

  6. Divide by −15-15: w=1915w = \frac{19}{15}

Final answer

w = 1915\frac{19}{15}

NEXT STEPS
Final Takeaways from Your Teacher

  • Review the property of equality: any operation applied to one side must distribute to every term on that side.

  • Practice drawing brackets around whole terms before clearing denominators in multi-term linear equations.

Lock it in

Question

How do you correctly expand the expression −(a−b)-(a - b)?

Answer

−a+b-a + b. A negative sign in front of parentheses acts as multiplying every term inside by −1-1.

Similar question

01

Solve the linear equation for xx: 7−(2x−3)=167 - (2x - 3) = 16.

Answer

x=−3x = -3. Key steps: Distribute the negative sign to get 7−2x+3=16  ⟹  10−2x=16  ⟹  −2x=6  ⟹  x=−37 - 2x + 3 = 16 \implies 10 - 2x = 16 \implies -2x = 6 \implies x = -3.

02

Solve the linear equation for yy: 4−(3y−5)=2−y−124 - (3y - 5) = 2 - \frac{y - 1}{2}.

Answer

y=3y = 3. Key steps: Simplify the left side to 9−3y9 - 3y. Multiply both sides by 22: 2(9−3y)=4−(y−1)  ⟹  18−6w=4−y+1=5−y  ⟹  −5y=−13  ⟹  y=1352(9 - 3y) = 4 - (y - 1) \implies 18 - 6w = 4 - y + 1 = 5 - y \implies -5y = -13 \implies y = \frac{13}{5}.

03

Solve the linear equation for mm: 3−2(3−2m)5=1−(4m−7)3 - \frac{2(3 - 2m)}{5} = 1 - (4m - 7).

Answer

m=2824=76m = \frac{28}{24} = \frac{7}{6}. Key steps: Expand the right side to 1−4m+7=8−4m1 - 4m + 7 = 8 - 4m. Multiply both sides by 55: 15−2(3−2m)=5(8−4m)  ⟹  15−6+4m=40−20m  ⟹  9+4m=40−20m  ⟹  24m=31  ⟹  m=312415 - 2(3 - 2m) = 5(8 - 4m) \implies 15 - 6 + 4m = 40 - 20m \implies 9 + 4m = 40 - 20m \implies 24m = 31 \implies m = \frac{31}{24}.

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