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Solve the linear equation for ww, paying strict attention to directional negative signs and distributive operations: −(4−3w)−5(2−w)=−2(w+7)−(6−4w)-(4 - 3w) - 5(2 - w) = -2(w + 7) - (6 - 4w).

Answer and steps

Final answer

w=−1w = -1

Steps

01

Expand both sides using the distributive property

To solve the linear equation, systematically expand both sides using the distributive property, combine like terms, and then isolate the variable ww.

Be especially careful with negative signs when distributing across parentheses:

  • A negative sign in front of parentheses reverses the sign of every term inside: −(a−b)=−a+b-(a - b) = -a + b.
  • Multiplying by a negative factor reverses the signs: −c(a−b)=−ca+cb-c(a - b) = -ca + cb.

Left-hand side: −(4−3w)−5(2−w)-(4 - 3w) - 5(2 - w)

  • Distribute the negative sign to (4−3w)(4 - 3w): −(4−3w)=−4+3w-(4 - 3w) = -4 + 3w
  • Distribute −5-5 to (2−w)(2 - w): −5(2−w)=−10+5w-5(2 - w) = -10 + 5w
  • Combine these parts: (−4+3w)+(−10+5w)=(3w+5w)+(−4−10)=8w−14(-4 + 3w) + (-10 + 5w) = (3w + 5w) + (-4 - 10) = 8w - 14

Right-hand side: −2(w+7)−(6−4w)-2(w + 7) - (6 - 4w)

  • Distribute −2-2 to (w+7)(w + 7): −2(w+7)=−2w−14-2(w + 7) = -2w - 14
  • Distribute the negative sign to (6−4w)(6 - 4w): −(6−4w)=−6+4w-(6 - 4w) = -6 + 4w
  • Combine these parts: (−2w−14)+(−6+4w)=(−2w+4w)+(−14−6)=2w−20(-2w - 14) + (-6 + 4w) = (-2w + 4w) + (-14 - 6) = 2w - 20
02

Equate the simplified expressions

8w−14=2w−208w - 14 = 2w - 20

03

Isolate the variable term on one side

Subtract 2w2w from both sides to collect the variable terms on the left: 8w−2w−14=−208w - 2w - 14 = -20 6w−14=−206w - 14 = -20

Add 1414 to both sides to isolate the variable term: 6w=−20+146w = -20 + 14 6w=−66w = -6

04

Solve for ww

Divide both sides by 66: w=−66=−1w = \frac{-6}{6} = -1

Common Pitfall to Avoid

When a minus sign precedes parentheses, such as −(4−3w)-(4 - 3w) or −(6−4w)-(6 - 4w), remember that the subtraction distributes to both terms inside. Writing −(4−3w)=−4−3w-(4 - 3w) = -4 - 3w instead of −4+3w-4 + 3w is a frequent sign error that alters the entire result.

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student forgets to distribute the leading negative sign to the second term inside the first set of parentheses, treating -(4 - 3w) as -4 - 3w, 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

    -(4 - 3w) - 5(2 - w) = -2(w + 7) - (6 - 4w)   ⟹  \implies -4 - 3w - 10 + 5w = -2w - 14 - 6 + 4w

    Correct

    -4 + 3w - 10 + 5w = -2w - 14 - 6 + 4w

    When expanding −(4−3w)-(4 - 3w), the negative sign must distribute to both terms inside the parentheses: −(4)−(−3w)=−4+3w-(4) - (-3w) = -4 + 3w. You left the second term as −3w-3w.

  2. Step 2

    2w - 14 = 2w - 20   ⟹  \implies 2w - 14 = -20 + 2w

    Correct

    8w - 14 = 2w - 20

    You correctly combined the terms from your previous line (−3w+5w=2w-3w + 5w = 2w), but because of the sign error in the first step, both sides ended up with 2w2w, leaving an apparent contradiction rather than a solvable linear equation.

WHAT YOU DID WELL
Compliments from Your Teacher

You correctly distributed −5(2−w)-5(2 - w) into −10+5w-10 + 5w, taking care of the double negative.

You accurately distributed −2(w+7)-2(w + 7) into −2w−14-2w - 14 on the right-hand side.

You correctly distributed the minus sign across −(6−4w)-(6 - 4w) into −6+4w-6 + 4w.

You combined your simplified algebraic terms consistently on both sides.

THE ERROR
Where Your ACEpath Teacher Spotted the Mistake
Distributive sign error

You distributed the leading negative sign to 44 but forgot to flip the sign of −3w-3w to +3w+3w.

WHY IT HAPPENED
ACEpath Teacher's Diagnosis

Treating a leading minus sign before parentheses as merely attaching to the first term rather than multiplying every term inside by −1-1.

REASONING PATH
Your ACEpath teacher’s suggested approach
Negation as Multiplication by -1

  • −(a−b)=−1⋅(a−b)=−a+b-(a - b) = -1 \cdot (a - b) = -a + b

  • Whenever you see a minus sign directly before parentheses, write an explicit −1-1 in front to remind yourself to multiply each term inside by −1-1.

CORRECTED PATH
ACEpath Teacher's Demonstration

  1. Distribute correctly across all parentheses: (−4+3w)−10+5w=−2w−14−6+4w(-4 + 3w) - 10 + 5w = -2w - 14 - 6 + 4w

  2. Combine like terms on both sides: 8w−14=2w−208w - 14 = 2w - 20

  3. Subtract 2w2w from both sides: 6w−14=−206w - 14 = -20

  4. Add 1414 to both sides: 6w=−66w = -6

  5. Divide by 66: w=−1w = -1

Final answer

w = -1

NEXT STEPS
Final Takeaways from Your Teacher

  • Rewrite expressions of the form −(a−b)-(a - b) as −1(a−b)-1(a - b) before distributing.

  • Draw distribution arrows to each term inside parentheses to ensure every sign flips.

  • If variable terms cancel unexpectedly leaving an impossible statement, recheck your distributed signs first.

OverviewACEpath Teacher's Diagnosis

02 The student fails to distribute the negative sign when expanding -5(2 - w), writing -10 - 5w instead of -10 + 5w, leading to -2w - 14 = 2w - 20 and w = 3/2.

Sample Student Work

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

    -4 + 3w - 10 - 5w = -2w - 14 - 6 + 4w

    Correct

    -4 + 3w - 10 + 5w = -2w - 14 - 6 + 4w

    When distributing −5-5 into (2−w)(2 - w), multiplying the negative constant by the subtracted variable gives (−5)×(−w)=+5w(-5) \times (-w) = +5w. You wrote −5w-5w instead of +5w+5w.

  2. Step 2

    -2w - 14 = 2w - 20

    Correct

    8w - 14 = 2w - 20

    You correctly combined like terms based on your previous line, but the result was affected by the earlier −5w-5w sign error.

  3. Step 3

    -4w - 14 = -20   ⟹  \implies -4w = -6

    Correct

    6w - 14 = -20   ⟹  \implies 6w = -6

    Your algebraic manipulations of subtracting 2w2w and adding 1414 were mathematically valid, but carried forward the incorrect variable coefficient.

  4. Step 4

    w = 32\frac{3}{2}

    Correct

    w = -1

    You divided correctly by −4-4 to isolate ww, but the propagated error led to 32\frac{3}{2} instead of −1-1.

WHAT YOU DID WELL
Compliments from Your Teacher

You correctly distributed the negative sign across −(4−3w)-(4 - 3w) to obtain −4+3w-4 + 3w.

You correctly expanded the right-hand side expressions −2(w+7)-2(w + 7) and −(6−4w)-(6 - 4w).

You maintained sound equation-balancing steps throughout the rest of your solution.

THE ERROR
Where Your ACEpath Teacher Spotted the Mistake
Distributive Sign Error

You treated −5(2−w)-5(2 - w) as −10−5w-10 - 5w instead of −10+5w-10 + 5w, losing track of the fact that multiplying two negative signs yields a positive term.

WHY IT HAPPENED
ACEpath Teacher's Diagnosis

When expanding parentheses preceded by a negative factor and containing an internal subtraction, the interaction between the outer negative and the inner minus sign was overlooked.

REASONING PATH
Your ACEpath teacher’s suggested approach
Double Negative in Distribution

  • −a(b−c)=−ab+ac-a(b - c) = -ab + ac

  • Always treat subtraction inside parentheses as addition of a negative number: −5(2−w)=−5(2+(−w))=−10+5w-5(2 - w) = -5(2 + (-w)) = -10 + 5w.

CORRECTED PATH
ACEpath Teacher's Demonstration

  1. Correctly expand both sides: −4+3w−10+5w=−2w−14−6+4w-4 + 3w - 10 + 5w = -2w - 14 - 6 + 4w

  2. Combine like terms on each side: 8w−14=2w−208w - 14 = 2w - 20

  3. Subtract 2w2w from both sides: 6w−14=−206w - 14 = -20

  4. Add 1414 to both sides: 6w=−66w = -6

  5. Divide by 66: w=−1w = -1

Final answer

w = -1

NEXT STEPS
Final Takeaways from Your Teacher

  • Before combining like terms, explicitly circle each multiplier and the sign of each term inside parentheses to ensure double negatives turn into addition.

  • Substitute your final solution back into the original equation to verify that both sides evaluate to the same value.

OverviewACEpath Teacher's Diagnosis

03 The student fails to distribute the negative sign to the second term in -(6 - 4w) on the right side, writing -6 - 4w instead of -6 + 4w, which yields 8w - 14 = -6w - 20 and w = -3/7.

Sample Student Work

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

    -4 + 3w - 10 + 5w = -2w - 14 - 6 - 4w

    Correct

    -4 + 3w - 10 + 5w = -2w - 14 - 6 + 4w

    When distributing the negative sign across (6−4w)(6 - 4w), remember that −(a−b)=−a+b-(a - b) = -a + b. Distributing −1-1 across −4w-4w yields +4w+4w, not −4w-4w.

  2. Step 2

    8w - 14 = -6w - 20

    Correct

    8w - 14 = 2w - 20

    You correctly combined like terms based on your previous line, but the right-hand side was affected by the earlier sign error.

  3. Step 3

    14w - 14 = -20   ⟹  \implies 14w = -6

    Correct

    6w - 14 = -20   ⟹  \implies 6w = -6

    Your steps to isolate the variable term were performed correctly based on your intermediate equation.

  4. Step 4

    w = -37\frac{3}{7}

    Correct

    w = -1

    You simplified the fraction correctly from your previous step, but the final value is incorrect due to the sign error in step 1.

WHAT YOU DID WELL
Compliments from Your Teacher

You correctly distributed the negative sign and coefficients across −(4−3w)-(4 - 3w), −5(2−w)-5(2 - w), and −2(w+7)-2(w + 7).

You combined like terms systematically on both sides of the equation.

You executed the properties of equality accurately to isolate the variable.

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

You treated −(6−4w)-(6 - 4w) as −6−4w-6 - 4w instead of flipping the sign of −4w-4w to +4w+4w.

WHY IT HAPPENED
ACEpath Teacher's Diagnosis

Incomplete distribution of the leading negative sign across parentheses containing a subtracted term, treating the minus sign in front of the parentheses as only attaching to the leading term.

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

  • −(a−b)=−1⋅a−(−1)⋅b=−a+b-(a - b) = -1 \cdot a - (-1) \cdot b = -a + b

  • Draw arrows from the outside minus sign or write an explicit −1-1 in front of the parentheses to make sure both terms change their signs.

CORRECTED PATH
ACEpath Teacher's Demonstration

  1. Expand the right-hand side with the correct sign: −2(w+7)−(6−4w)=−2w−14−6+4w-2(w + 7) - (6 - 4w) = -2w - 14 - 6 + 4w.

  2. Combine like terms on both sides: 8w−14=2w−208w - 14 = 2w - 20.

  3. Subtract 2w2w from both sides: 6w−14=−206w - 14 = -20.

  4. Add 1414 to both sides: 6w=−66w = -6.

  5. Divide by 66: w=−1w = -1.

Final answer

w = -1

NEXT STEPS
Final Takeaways from Your Teacher

  • Review the rule for distributing negative factors over subtraction: −1⋅(−b)=+b-1 \cdot (-b) = +b.

  • When expanding expressions like −(a−b)-(a - b), rewrite them as −1(a−b)-1(a - b) as an intermediate check.

  • Try the practice exercises below to solidify distributing negative signs.

Lock it in

Question

What does a negative sign immediately in front of parentheses do to the signs inside, such as in −(a−b)-(a - b)?

Answer

It reverses the sign of every term inside: −(a−b)=−a+b-(a - b) = -a + b because you are multiplying by −1-1.

Similar question

01

Solve for xx: −(3−2x)+4=15-(3 - 2x) + 4 = 15.

Answer

x=7x = 7. Distribute the negative: −3+2x+4=15  ⟹  2x+1=15  ⟹  2x=14  ⟹  x=7-3 + 2x + 4 = 15 \implies 2x + 1 = 15 \implies 2x = 14 \implies x = 7.

02

Solve for yy: 2(y−3)−(5−4y)=3(y+1)+22(y - 3) - (5 - 4y) = 3(y + 1) + 2.

Answer

y=5y = 5. Expand both sides: 2y−6−5+4y=3y+3+2  ⟹  6y−11=3y+5  ⟹  3y=16  ⟹  y=1632y - 6 - 5 + 4y = 3y + 3 + 2 \implies 6y - 11 = 3y + 5 \implies 3y = 16 \implies y = \frac{16}{3}.

03

Solve for mm: −(2−5m)−3(4−m)=−4(1−2m)−10-(2 - 5m) - 3(4 - m) = -4(1 - 2m) - 10.

Answer

No solution. Expand: −2+5m−12+3m=−4+8m−10  ⟹  8m−14=8m−14-2 + 5m - 12 + 3m = -4 + 8m - 10 \implies 8m - 14 = 8m - 14. Subtracting 8m8m gives −14=−14-14 = -14, which means the equation is an identity with infinitely many solutions (all real numbers satisfy it).

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