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Solve the following linear equation for xx: 7(2x−5)−3(x+4)=4(x−8)+117(2x - 5) - 3(x + 4) = 4(x - 8) + 11.

Answer and steps

Final answer

x=4x = 4

Steps

01

Expand both sides using the distributive property

To solve the linear equation, follow the standard procedure of expanding, combining like terms, isolating the variable, and solving.

Multiply each term inside the parentheses by the factor outside:

  • On the left-hand side: 7(2x−5)−3(x+4)=7⋅2x−7⋅5−3⋅x−3⋅47(2x - 5) - 3(x + 4) = 7 \cdot 2x - 7 \cdot 5 - 3 \cdot x - 3 \cdot 4 =14x−35−3x−12= 14x - 35 - 3x - 12

    Common Pitfall: When distributing a negative number such as −3-3 across (x+4)(x + 4), be sure to multiply both terms by −3-3, resulting in −3x−12-3x - 12, not −3x+12-3x + 12.

  • On the right-hand side: 4(x−8)+11=4⋅x−4⋅8+114(x - 8) + 11 = 4 \cdot x - 4 \cdot 8 + 11 =4x−32+11= 4x - 32 + 11
02

Combine like terms on each side

Group the variable terms and constant terms together:

  • Left-hand side: (14x−3x)+(−35−12)=11x−47(14x - 3x) + (-35 - 12) = 11x - 47
  • Right-hand side: 4x+(−32+11)=4x−194x + (-32 + 11) = 4x - 19

The equation simplifies to: 11x−47=4x−1911x - 47 = 4x - 19

03

Isolate the variable terms on one side

Subtract 4x4x from both sides to collect all terms containing xx on the left: 11x−4x−47=−1911x - 4x - 47 = -19 7x−47=−197x - 47 = -19

04

Isolate the constant terms on the other side

Add 4747 to both sides of the equation: 7x=−19+477x = -19 + 47 7x=287x = 28

05

Solve for xx

Divide both sides by the coefficient of xx, which is 77: x=287x = \frac{28}{7} x=4x = 4

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 when expanding -3(x + 4), resulting in x = 50/7.

Sample Student Work

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

    14x−35−3x+12=4x−32+1114x - 35 - 3x + 12 = 4x - 32 + 11

    Correct

    14x−35−3x−12=4x−32+1114x - 35 - 3x - 12 = 4x - 32 + 11

    When distributing −3-3 across (x+4)(x + 4), multiplying the negative factor −3-3 by +4+4 results in −12-12, not +12+12.

  2. Step 2

    11x−23=4x−2111x - 23 = 4x - 21

    Correct

    11x−47=4x−2111x - 47 = 4x - 21

    You combined like terms correctly on both sides according to your expanded equation, but the left-hand constant is affected by the earlier sign error.

  3. Step 3

    7x=27x = 2

    Correct

    7x=267x = 26

    You correctly subtracted 4x4x and added 2323 across the equals sign, but the constant value carries forward the earlier error.

  4. Step 4

    x=27x = \frac{2}{7}

    Correct

    x=267x = \frac{26}{7}

    You correctly divided both sides by 77, but the final answer is affected by the propagated error.

WHAT YOU DID WELL
Compliments from Your Teacher

You correctly expanded the positive terms 7(2x−5)7(2x - 5) and 4(x−8)4(x - 8).

You combined like terms accurately on the right-hand side to obtain 4x−214x - 21.

You followed standard equation-solving procedures to isolate the variable and divide by its coefficient.

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

You wrote +12+12 instead of −12-12 when expanding −3(x+4)-3(x + 4).

WHY IT HAPPENED
ACEpath Teacher's Diagnosis

When expanding a term preceded by a minus sign, such as −a(b+c)-a(b + c), the negative sign must be distributed to every term inside the parentheses, giving −ab−ac-ab - ac.

REASONING PATH
Your ACEpath teacher’s suggested approach
Distribute the Negative Sign to Every Term

  • For any expression −a(b+c)-a(b + c), the distributive property gives (−a)⋅b+(−a)⋅c=−ab−ac(-a) \cdot b + (-a) \cdot c = -ab - ac.

  • Circle the negative coefficient together with its minus sign before multiplying each term inside the parentheses.

CORRECTED PATH
ACEpath Teacher's Demonstration

  1. Expand both sides carefully with correct signs: 14x−35−3x−12=4x−32+1114x - 35 - 3x - 12 = 4x - 32 + 11.

  2. Combine like terms on each side: 11x−47=4x−2111x - 47 = 4x - 21.

  3. Subtract 4x4x from both sides: 7x−47=−217x - 47 = -21.

  4. Add 4747 to both sides: 7x=267x = 26.

Final answer

x=267x = \frac{26}{7}

NEXT STEPS
Final Takeaways from Your Teacher

  • Circle or highlight negative signs in front of parentheses before distributing.

  • Write out the intermediate product, such as (−3)⋅(+4)=−12(-3) \cdot (+4) = -12, to prevent sign slips.

  • Practice distributing negative constants across expressions with mixed addition and subtraction.

OverviewACEpath Teacher's Diagnosis

02 The student forgets to multiply the second term inside the parentheses by the outside factor, writing -3(x + 4) as -3x - 4.

Sample Student Work

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

    14x - 35 - 3x - 4 = 4x - 32 + 11

    Correct

    14x - 35 - 3x - 12 = 4x - 32 + 11

    You correctly expanded 7(2x−5)=14x−357(2x - 5) = 14x - 35 and 4(x−8)=4x−324(x - 8) = 4x - 32. However, for −3(x+4)-3(x + 4), the factor −3-3 must be multiplied by every term inside the parentheses: −3⋅x+(−3)⋅4=−3x−12-3 \cdot x + (-3) \cdot 4 = -3x - 12, not −3x−4-3x - 4.

  2. Step 2

    11x - 39 = 4x - 21

    Correct

    11x - 47 = 4x - 19

    You combined like terms correctly according to your previous line ((14x−3x)+(−35−4)=11x−39(14x - 3x) + (-35 - 4) = 11x - 39), but the constant term on the left was affected by the earlier distribution error.

  3. Step 3

    7x = 18

    Correct

    7x = 28

    Your algebraic isolation technique was correct: you subtracted 4x4x from both sides (11x−4x=7x11x - 4x = 7x) and added 3939 to −21-21 to get 1818, though the values were carried over from the earlier error.

  4. Step 4

    x = 187\frac{18}{7}

    Correct

    x = 4

    You correctly divided both sides by 77, but the final numerical value reflects the earlier expansion mistake.

WHAT YOU DID WELL
Compliments from Your Teacher

You correctly expanded 7(2x−5)7(2x - 5) into 14x−3514x - 35 and 4(x−8)4(x - 8) into 4x−324x - 32.

You combined like terms and balanced both sides of the linear equation with sound algebraic operations.

THE ERROR
Where Your ACEpath Teacher Spotted the Mistake
Incomplete distribution across parentheses

You distributed −3-3 to the variable xx but copied down −4-4 without multiplying 44 by −3-3.

WHY IT HAPPENED
ACEpath Teacher's Diagnosis

Failing to distribute the outside factor to all terms inside the parentheses, particularly when the outside factor is negative.

REASONING PATH
Your ACEpath teacher’s suggested approach
Distribute to All Terms

  • a(b+c)=a⋅b+a⋅ca(b + c) = a \cdot b + a \cdot c

  • Draw multiplication arrows from the outside multiplier to every single term inside the parentheses before writing the expanded terms.

Include the Sign with the Factor

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

  • Treat the minus sign as attached to the number: multiply both xx and +4+4 by −3-3 to get −3x−12-3x - 12.

CORRECTED PATH
ACEpath Teacher's Demonstration

  1. Expand left and right sides: 14x−35−3x−12=4x−32+1114x - 35 - 3x - 12 = 4x - 32 + 11

  2. Combine like terms on each side: 11x−47=4x−1911x - 47 = 4x - 19

  3. Subtract 4x4x from both sides: 7x−47=−197x - 47 = -19

  4. Add 4747 to both sides: 7x=287x = 28

  5. Divide by 77: x=4x = 4

Final answer

x=4x = 4

NEXT STEPS
Final Takeaways from Your Teacher

  • Practice drawing distribution arrows from the outside coefficient to each internal term before multiplying.

  • Double-check every negative multiplier by expanding it explicitly: write (−3)(x)+(−3)(4)(-3)(x) + (-3)(4) before simplifying.

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

OverviewACEpath Teacher's Diagnosis

03 The student makes a sign error when transposing constant terms across the equals sign, adding 47 as negative instead of positive.

Sample Student Work

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

    14x - 35 - 3x - 12 = 4x - 32 + 11

  2. Step 2

    11x - 47 = 4x - 21

  3. Step 3

    7x = -21 - 47

    Correct

    7x = -21 + 47   ⟹  \implies 7x = 26

    To eliminate −47-47 from the left side, you must add 4747 to both sides using the addition property of equality. Writing −21−47-21 - 47 subtracted 4747 instead of adding it.

  4. Step 4

    7x = -68   ⟹  \implies x = -687\frac{68}{7}

    Correct

    x = 267\frac{26}{7}

    Your arithmetic −21−47=−68-21 - 47 = -68 and subsequent division by 77 were logically followed through, but the result was altered by the sign error in the previous step.

WHAT YOU DID WELL
Compliments from Your Teacher

You distributed the factors 77, −3-3, and 44 across their respective parentheses accurately.

You combined like terms on each side of the equation without making arithmetic errors.

You isolated the variable term 7x7x correctly by subtracting 4x4x from both sides.

THE ERROR
Where Your ACEpath Teacher Spotted the Mistake
Sign error during term transposition

You kept the negative sign on 4747 when shifting it to the right-hand side instead of changing it to +47+47.

WHY IT HAPPENED
ACEpath Teacher's Diagnosis

Treating transposition as simply moving a number to the other side rather than applying the inverse operation (adding 4747 to both sides) to maintain equality.

REASONING PATH
Your ACEpath teacher’s suggested approach
Inverse Operation Check

  • To cancel a subtracted term −c-c on one side of an equation, you must add +c+c to both sides.

  • Always write the balancing step explicitly: 11x−47+47=4x−21+4711x - 47 + 47 = 4x - 21 + 47.

Transposition Sign Rule

  • Whenever any term crosses the equals sign, its sign must flip.

  • Check every transposed term immediately: a negative term on the left must appear with a positive sign on the right.

CORRECTED PATH
ACEpath Teacher's Demonstration

  1. Add 4747 to both sides of the equation to isolate the variable term: 7x=−21+477x = -21 + 47.

  2. Compute the right-hand side sum: 7x=267x = 26.

  3. Divide both sides by 77: x=267x = \frac{26}{7}.

Final answer

x = 267\frac{26}{7}

NEXT STEPS
Final Takeaways from Your Teacher

  • Practice writing the balancing operation explicitly under both sides (e.g., +47+47 under each side) before combining.

  • Review the golden rule of transposition: crossing the equals sign always flips the term's sign.

  • Complete the targeted flashcards and practice problems below to master isolating terms across the equals sign.

Lock it in

Question

What is the general expansion rule for −a(b+c)-a(b + c)?

Answer

−ab−ac-ab - ac. The negative sign applies to every term inside the parentheses, reversing each term's sign.

Similar question

01

Solve the linear equation for xx: 3(x−2)−2(x+5)=13(x - 2) - 2(x + 5) = 1.

Answer

x=17x = 17. Expanding gives 3x−6−2x−10=13x - 6 - 2x - 10 = 1. Combining like terms yields x−16=1x - 16 = 1, so x=17x = 17.

02

Solve the linear equation for yy: 5(2y−1)−4(3y+2)=2(y−7)+55(2y - 1) - 4(3y + 2) = 2(y - 7) + 5.

Answer

y=−1y = -1. Expanding gives 10y−5−12y−8=2y−14+510y - 5 - 12y - 8 = 2y - 14 + 5, which simplifies to −2y−13=2y−9-2y - 13 = 2y - 9. Rearranging terms gives −4y=4-4y = 4, so y=−1y = -1.

03

Solve the linear equation for tt: 2−3(4t−1)−(2t+5)=−5(3t−2)2 - 3(4t - 1) - (2t + 5) = -5(3t - 2).

Answer

t=10t = 10. Expanding gives 2−12t+3−2t−5=−15t+102 - 12t + 3 - 2t - 5 = -15t + 10, which simplifies to −14t=−15t+10-14t = -15t + 10. Adding 15t15t to both sides yields t=10t = 10.

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