Solve each linear equation. See Examples 1–3. -4 (2x - 6) + 8x = 5x + 24 + x

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First, apply the distributive property to the term \(-4 (2x - 6)\). This means multiplying \(-4\) by each term inside the parentheses: \(-4 \times 2x\) and \(-4 \times -6\).

Rewrite the equation after distribution: \(-8x + 24 + 8x = 5x + 24 + x\).

Combine like terms on the left side: \(-8x + 8x\) simplifies to \(0\), so the left side becomes \(24\). On the right side, combine \(5x + x\) to get \(6x\). Now the equation is \(24 = 6x + 24\).

Next, isolate the variable term by subtracting \(24\) from both sides: \(24 - 24 = 6x + 24 - 24\), which simplifies to \(0 = 6x\).

Finally, solve for \(x\) by dividing both sides by \(6\): \(\frac{0}{6} = \frac{6x}{6}\), which simplifies to \(x = 0\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Distributive Property

The distributive property allows you to multiply a single term outside the parentheses by each term inside the parentheses. For example, -4(2x - 6) becomes -8x + 24. This step is essential to simplify expressions and combine like terms.

Combining Like Terms

Combining like terms involves adding or subtracting terms that have the same variable raised to the same power. For instance, 8x and 5x can be combined to 13x. This simplifies the equation and makes it easier to isolate the variable.

Solving Linear Equations

Solving linear equations means finding the value of the variable that makes the equation true. This involves isolating the variable on one side by performing inverse operations such as addition, subtraction, multiplication, or division.

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