Q. \( -10x + 6(-x + 3) = -(6x – 6) – 7x \)

Answer

Solve
\[
-10x+6(-x+3)=-(6x-6)-7x.
\]
Expand both sides:
\[
-10x-6x+18=-6x+6-7x \implies -16x+18=-13x+6.
\]
Bring x-terms together:
\[
-16x+13x=6-18 \implies -3x=-12 \implies x=4.
\]

Final answer: \(x=4\).

Detailed Explanation

Solve the equation step-by-step

Write the original equation:
\[ -10x + 6(-x + 3) = -(6x – 6) – 7x \]
Expand the parentheses on each side. On the left, multiply 6 by each term in \(-x + 3\). On the right, distribute the leading negative sign through \(6x – 6\).
\[ -10x + 6(-x) + 6(3) = -1(6x) + -1(-6) – 7x \]
\[ -10x – 6x + 18 = -6x + 6 – 7x \]
Combine like terms on both sides. Combine the x-terms on each side.
\[ (-10x – 6x) + 18 = (-6x – 7x) + 6 \]
\[ -16x + 18 = -13x + 6 \]
Isolate the variable terms. Move the x-terms to one side by adding \(16x\) to both sides (or adding \(13x\) to both sides; here we add \(16x\)).
\[ 18 = (-13x + 16x) + 6 \]
\[ 18 = 3x + 6 \]
Solve for \(x\). Subtract 6 from both sides, then divide by 3.
\[ 18 – 6 = 3x \]
\[ 12 = 3x \]
\[ x = 4 \]
Check the solution by substitution. Substitute \(x = 4\) into the original equation to verify both sides are equal.
Left:
\[ -10(4) + 6(-4 + 3) = -40 + 6(-1) = -40 – 6 = -46 \]
Right:
\[ -(6(4) – 6) – 7(4) = -(24 – 6) – 28 = -18 – 28 = -46 \]
Both sides equal \(-46\), so the solution is correct.

Solution: \( x = 4 \)

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FAQs

Q1: What is the first step to simplify \( -10x+6(-x+3)=-(6x-6)-7x \)?

A1: Distribute: \(6(-x+3)=-6x+18\) and \(-(6x-6)=-6x+6\). Then rewrite the equation as \( -10x-6x+18=-6x+6-7x\).

Q2: How do I combine like terms?

A2: Combine on each side: LHS \( -10x-6x+18=-16x+18\). RHS \( -6x-7x+6=-13x+6\). So you get \( -16x+18=-13x+6\).

Q3: How do I isolate \(x\)?

A3: Move \(x\)-terms and constants: \( -16x+18=-13x+6\) ⇒ \( -16x+13x=6-18\) ⇒ \( -3x=-12\) ⇒ \( x=4\).

Q4: How do I check the solution \(x=4\)?

A4: Substitute: LHS \( -10(4)+6(-4+3)=-40+6(-1)=-46\). RHS \( -(6·4-6)-7·4=-(24-6)-28=-18-28=-46\). Both sides equal, so \(x=4\) is correct.

Q5: What common sign errors should I watch for?

A5: Don’t forget to distribute negatives across parentheses: \(-(6x-6)=-6x+6\). -Also combine like terms carefully (e.g., \(-10x-6x=-16x\)), and keep track when moving terms across the equals sign.

Q6: What if variables cancel when simplifying?

A6: If all \(x\)-terms cancel and you get a true statement (e.g., \(18=18\)), there are infinitely many solutions. If you get a false statement (e.g., \(5=2\)), there is no solution.

Q7: Can I rearrange steps to solve faster?

A7: Yes - simplify both sides fully first, then collect variable terms on one side and constants on the other. That minimizes algebraic errors and often reduces arithmetic.

Q8: How to avoid arithmetic mistakes when combining terms?

A8: Write each simplification step separately, check distribution, align like terms vertically, and recompute the final numeric combination to verify consistency.

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