Problem 56 - Elementary Algebra

Question

Solve. $$ 12-2(2 x+1)=4(x-1) $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{7}{4}\).

Step 1: Expand the Expressions

First, we need to distribute the terms within the parentheses. On the left side, distribute \(-2\) across \((2x+1)\), and on the right side, distribute \(4\) across \((x-1)\). This gives us: \[ 12 - 2(2x) - 2(1) = 4x - 4 \] Simplifying, we get: \[ 12 - 4x - 2 = 4x - 4 \] Combine like terms: \[ 10 - 4x = 4x - 4 \]

Step 2: Move Variables to One Side

Now, move all the \(x\) terms to one side of the equation. Add \(4x\) to both sides to eliminate \(-4x\) from the left side: \[ 10 - 4x + 4x = 4x + 4x - 4 \] Which simplifies to: \[ 10 = 8x - 4 \]

Step 3: Isolate the Variable

Next, isolate \(x\) by moving constant terms to the other side of the equation. Add \(4\) to both sides so that:\[ 10 + 4 = 8x \] This gives: \[ 14 = 8x \]

Step 4: Solve for x

Finally, divide both sides by \(8\) to solve for \(x\):\[ \frac{14}{8} = x \] Simplify the fraction: \[ \frac{7}{4} = x \] So, the solution is: \[ x = \frac{7}{4} \]

Key Concepts

Distributive Property

The distributive property is a key algebraic rule that helps simplify expressions and solve equations. It allows us to multiply a single term by each term inside a set of parentheses. In this exercise, we see the distributive property at work on both sides of the equation:

• Distributing \(-2\) to \(2x+1\) gives us \-2(2x) - 2(1)\.
• Similarly, distributing \(4\) to \(x-1\) results in \4x - 4\.

After using the distributive property, the expression transforms from a complex combination into a simpler one, which we can work with to further solve the equation more easily. Always remember, the distributive property is useful when dealing with expressions where terms are grouped in parentheses.

Combining Like Terms

Once expressions have been expanded using the distributive property, the next step is to simplify by combining like terms. "Like terms" have the same variable raised to the same power, or are constants.
Take the equation from the exercise after applying the distributive property: \[ 12 - 4x - 2 = 4x - 4 \]

• Combine the constants on the left: \12 - 2\ results in \10\.
• The terms containing \(x\) are \-4x\ on the left and \4x\ on the right. There's no combining to do here, but repositioning them is important.

Now the equation looks much simpler: \[ 10 - 4x = 4x - 4 \] Combining like terms simplifies equations, making them easier to solve.

Solving Linear Equations

To solve linear equations, our goal is to isolate the variable, in this case, \(x\), on one side of the equation. Let's walk through it:

• Start by moving all \(x\) terms to one side. Add \4x\ to both sides, so the equation reads \10 = 8x - 4\.
• Next, shift the constants to the opposite side by adding \4\ to both sides: \10 + 4 = 8x\, simplifying to \14 = 8x\.
• To isolate \(x\), divide both sides by \8\: \x = \frac{14}{8}\.

Further simplifying the fraction, \x = \frac{7}{4}\. When solving linear equations, systematic rearranging and isolation of the variable help us find the solution efficiently. Breaking down each step ensures clarity throughout the solving process.