Problem 28 - Prealgebra

Question

Solve each equation using the methods shown in this section. $$8 x-10=-4(x+3)+2$$

Step-by-Step Solution

Answer

The solution is \( x = 0 \).

Step 1: Distribute on the Right Side

To eliminate the parentheses on the right side of the equation, we need to distribute the -4 across the terms inside the parentheses. This means multiplying -4 by both x and 3.The equation becomes: \[ 8x - 10 = -4x - 12 + 2 \]

Step 2: Simplify the Right Side

Combine like terms on the right side of the equation. Add -12 and 2 together: \[ -4x - 12 + 2 = -4x - 10 \] Now our equation is: \[ 8x - 10 = -4x - 10 \]

Step 3: Move All Variable Terms to One Side

Add 4x to both sides of the equation to get all x terms on one side: \[ 8x + 4x - 10 = -10 \] Simplify the left side to: \[ 12x - 10 = -10 \]

Step 4: Simplify by Moving Constant Terms

Add 10 to both sides of the equation to isolate the term with x: \[ 12x - 10 + 10 = -10 + 10 \] This simplifies to: \[ 12x = 0 \]

Step 5: Divide to Solve for x

Divide both sides by 12 to solve for x: \[ x = \frac{0}{12} \] Simplifying gives us: \[ x = 0 \]

Key Concepts

Distributive Property

The distributive property is a valuable tool in algebra that allows us to eliminate parentheses. It helps in spreading out a multiplication operation over terms inside the parenthesis. Imagine you have a parenthesis situation like \(-4(x + 3)\). To get rid of the parentheses, you distribute \(-4\) across the terms inside, meaning you multiply \(-4\) by each element within the parentheses:

• Multiply \(-4\) by \(x\), which gives \(-4x\).
• Multiply \(-4\) by \(3\), resulting in \(-12\).

Once distributed, our expression changes from \(-4(x + 3)\) to \(-4x - 12\). This property simplifies complex equations and makes them much easier to handle.
By understanding and applying the distributive property, the equation on the right side becomes free of parentheses, thus making subsequent steps like combining terms straightforward.

Combining Like Terms

After using the distributive property, you often end up with an equation that has terms which can be combined. This is where combining like terms comes in handy. Like terms have identical variables raised to the same power. You simply add or subtract the coefficients of these terms.

• In the equation \(-4x - 12 + 2\), the numbers \(-12\) and \(2\) are like terms because they are both constants.
• You add them together to get \(-10\).

Now the equation is \(8x - 10 = -4x - 10\), where \(-4x\) and \(8x\) can be considered like terms. But first, they must be on the same side to make it easier to combine. Combining like terms helps simplify equations, significantly reducing the number of terms you need to work with, and making it easier to solve for the variable.

Isolating Variables

Isolating the variable is often the final step in solving equations, as it leads directly to finding the value of the unknown. To isolate the variable means to get the variable on one side of the equation by itself. Here's how you do it:

• Add or subtract terms to move all variable terms to one side. For instance, adding \(4x\) to both sides of \(8x - 10 = -4x - 10\) gives \(12x - 10 = -10\).
• Then, rid the equation of constant terms surrounding the variable by adding or subtracting them. In this case, add \(10\) to both sides, leading to \(12x = 0\).
• Finally, divide by the coefficient of the variable. Dividing both sides by \(12\) results in \(x = 0\).

By systematically following these steps, you effectively isolate the variable. This approach transforms the equation into a simple statement where the variable equals a specific number, making it easy to interpret the solution.