Solve the following multi-step linear equation for the unknown variable $x$: $4(2x - 5) - 3(x - 4) = 5x - 2(3

Solve the following multi-step linear equation for the unknown variable $x$ : $4(2x - 5) - 3(x - 4) = 5x - 2(3 - x)$ .

Final answer

$x = -1$

Steps

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

Step 1: Apply the distributive property to eliminate all parentheses.

Multiply the factors outside each set of parentheses by every term inside:

On the left-hand side: $4(2x - 5) = 8x - 20$ $-3(x - 4) = -3x + 12$ (Take care with the negative sign: $-3 \cdot (-4) = +12$ .)
On the right-hand side: $-2(3 - x) = -6 + 2x$

Substitute these expanded terms back into the equation: $(8x - 20) - 3x + 12 = 5x - 6 + 2x$

Step 2: Combine like terms on each side.

Left-hand side: $(8x - 3x) + (-20 + 12) = 5x - 8$
Right-hand side: $(5x + 2x) - 6 = 7x - 6$

The simplified equation is: $5x - 8 = 7x - 6$

Step 3: Collect the variable terms on one side and the constant terms on the other.

Subtract $5x$ from both sides to keep the variable coefficient positive: $-8 = 7x - 5x - 6$ $-8 = 2x - 6$

Add $6$ to both sides to isolate the variable term: $-8 + 6 = 2x$ $-2 = 2x$

Step 4: Solve for $x$ .

Divide both sides by $2$ : $x = \frac{-2}{2} = -1$

Step 5: Verify the solution.

Substitute $x = -1$ back into the original equation:

Left-hand side: $4(2(-1) - 5) - 3((-1) - 4) = 4(-7) - 3(-5) = -28 + 15 = -13$
Right-hand side: $5(-1) - 2(3 - (-1)) = -5 - 2(4) = -5 - 8 = -13$

Since both sides equal $-13$ , the solution is verified.

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student forgets to distribute the negative sign to the second term inside parentheses when expanding -3(x - 4), writing -12 instead of +12.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
8x - 20 - 3x - 12 = 5x - 6 + 2x
Correct
8x - 20 - 3x + 12 = 5x - 6 + 2x
When expanding $-3(x - 4)$ , you must multiply $-3$ by both $x$ and $-4$ . Multiplying two negative numbers gives a positive result: $-3 \cdot (-4) = +12$ , not $-12$ .
Step 2
5x - 32 = 7x - 6
Correct
5x - 8 = 7x - 6
You correctly combined the like terms from your previous line ( $8x - 3x = 5x$ and $-20 - 12 = -32$ ), but this step is affected by the earlier sign error.
Step 3
-2x = 26
Correct
-2x = 2
Your equation solving mechanics were correct ( $5x - 7x = -2x$ and $-6 + 32 = 26$ ), though the numbers reflect the propagated error from Step 1.
Step 4
x = -13
Correct
x = -1
You accurately divided $26$ by $-2$ to get $-13$ , which follows logically from your previous line but differs from the correct solution $x = -1$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly expanded $4(2x - 5)$ into $8x - 20$ .

You correctly distributed $-2$ across $(3 - x)$ to obtain $-6 + 2x$ .

You accurately combined like terms and balanced the linear equation throughout all downstream steps.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Distributive Sign Error

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

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Treating the negative sign as an operation attached only to the first term rather than distributing the negative factor to every term inside the parentheses.

REASONING PATH

Your ACEpath teacher’s suggested approach

Distribute Negative Factors Completely: $-a(b - c) = -ab + ac$
Always circle or highlight the negative sign with the coefficient in front of parentheses before multiplying each interior term.

CORRECTED PATH

ACEpath Teacher's Demonstration

Expand the left-hand side: $4(2x - 5) - 3(x - 4) = 8x - 20 - 3x + 12$ .
Combine like terms on both sides: $5x - 8 = 7x - 6$ .
Subtract $7x$ from both sides: $-2x - 8 = -6$ .
Add $8$ to both sides: $-2x = 2$ .

Final answer

x = -1

NEXT STEPS

Final Takeaways from Your Teacher

Pay special attention to negative signs in front of parentheses, ensuring negative times negative yields a positive term.
Substitute your final answer back into both sides of the original equation to verify that they evaluate to the same value.

OverviewACEpath Teacher's Diagnosis

02 The student distributes -2 across (3 - x) as -6 - 2x instead of -6 + 2x by failing to multiply negative two by negative x.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
8x - 20 - 3x + 12 = 5x - 6 - 2x
Correct
8x - 20 - 3x + 12 = 5x - 6 + 2x
You correctly distributed $4$ and $-3$ on the left side, including getting $+12$ from $(-3) \cdot (-4)$ . However, on the right side, multiplying the negative factor $-2$ by $-x$ results in $+2x$ , not $-2x$ .
Step 2
5x - 8 = 3x - 6
Correct
5x - 8 = 7x - 6
You correctly combined $8x - 3x = 5x$ and $-20 + 12 = -8$ on the left, and followed your previous step on the right with $5x - 2x = 3x$ . Because of the earlier sign error, this step inherits the incorrect expression.
Step 3
2x = 2
Correct
-2x = 2 $\text{or}$ 2x = -2
Your transposition method of subtracting $3x$ and adding $8$ was algebraically valid based on your previous equation, but it carries forward the earlier error.
Step 4
x = 1
Correct
x = -1
You solved your linear equation correctly by dividing both sides by $2$ , but the final numerical value is incorrect due to the initial sign error.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly distributed $-3$ across $(x - 4)$ to obtain $-3x + 12$ .

You combined like linear and constant terms consistently on both sides of your equation.

You accurately isolated the variable and completed the final division step.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Distributive sign error with negative factors

You treated $-2(3 - x)$ as $-6 - 2x$ instead of $-6 + 2x$ , missing the sign flip that occurs when multiplying two negative values.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Inconsistent application of the sign rule during distribution: while you correctly handled $(-3) \cdot (-4) = +12$ on the left side, you lost track of the negative coefficient when distributing $-2$ across $-x$ on the right side.

REASONING PATH

Your ACEpath teacher’s suggested approach

Pair Each Sign with Its Term: Every minus sign belongs to the term directly following it: $-a(b - c) = (-a)(b) + (-a)(-c) = -ab + ac$ .
Before expanding, circle the negative multiplier along with its sign, and draw distribution arrows explicitly to each signed term inside the parentheses.
Check Signs Before Combining: Multiplying two like signs yields a positive product: $(\text{negative}) \times (\text{negative}) = \text{positive}$ .
Pause immediately after expanding parentheses to verify that each product of two negative terms is written with a plus sign.

CORRECTED PATH

ACEpath Teacher's Demonstration

Apply the distributive property on both sides: $8x - 20 - 3x + 12 = 5x - 6 + 2x$ .
Combine like terms on each side: $5x - 8 = 7x - 6$ .
Subtract $5x$ from both sides: $-8 = 2x - 6$ .
Add $6$ to both sides: $-2 = 2x$ .

Final answer

x = -1

NEXT STEPS

Final Takeaways from Your Teacher

Review the distributive property flashcards focusing on expanding negative coefficients across subtraction.
Solve the three targeted practice problems, checking your intermediate sign expansions at each step.
Practice checking your final answer by substituting it back into the original equation to verify that both sides balance.

OverviewACEpath Teacher's Diagnosis

03 The student forgets to change signs when transposing constant terms across the equals sign, adding 6 instead of subtracting it or confusing -8 + 6 with -8 - 6.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
8x - 20 - 3x + 12 = 5x - 6 + 2x $\implies$ 5x - 8 = 7x - 6
Step 2
5x - 7x = -6 - 8
Correct
5x - 7x = -6 + 8
When moving $-8$ from the left-hand side to the right-hand side, you must add $8$ to both sides, which yields $-6 + 8$ , not $-6 - 8$ .
Step 3
-2x = -14 $\implies$ x = 7
Correct
-2x = 2 $\implies$ x = -1
Your arithmetic and final division followed logically from your previous line, but the result was affected by the sign error in the constant term.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly applied the distributive property across all sets of parentheses.

You accurately combined like terms on both the left-hand and right-hand sides of the equation.

You maintained proper algebraic structure and divided cleanly by the variable's coefficient.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Sign error during term transposition

You copied the sign of $-8$ as $-8$ on the right side rather than performing the inverse operation $+8$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Treating transposition as simply moving a term across the equals sign without applying the inverse operation to keep both sides balanced.

REASONING PATH

Your ACEpath teacher’s suggested approach

Use Inverse Operations: Whatever operation is applied to one side must be balanced using the inverse operation.
To eliminate a subtracted term like $-8$ , add $8$ to both sides: $-8 + 8 = 0$ , so the other side becomes $-6 + 8$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Subtract $7x$ from both sides: $5x - 7x - 8 = -6$ , which simplifies to $-2x - 8 = -6$ .
Add $8$ to both sides to isolate $-2x$ : $-2x = -6 + 8$ , which gives $-2x = 2$ .
Divide both sides by $-2$ : $x = \frac{2}{-2} = -1$ .

Final answer

x = -1

NEXT STEPS

Final Takeaways from Your Teacher

Instead of moving terms mentally, explicitly write $+8$ on both sides of the equals sign to ensure balance.
Substitute your final answer back into both sides of the original equation to check if they evaluate to the same value.

Lock it in

Question

What is the expanded form of $-a(b - c)$ ?

Answer

$-ab + ac$ . A negative factor multiplied by a negative term produces a positive product.

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student forgets to distribute the negative sign to the second term inside parentheses when expanding -3(x - 4), writing -12 instead of +12.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

02 The student distributes -2 across (3 - x) as -6 - 2x instead of -6 + 2x by failing to multiply negative two by negative x.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

03 The student forgets to change signs when transposing constant terms across the equals sign, adding 6 instead of subtracting it or confusing -8 + 6 with -8 - 6.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

Lock it in

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