Problem 17 - Prealgebra

Question

Solve each equation using the methods shown in this section. $$2(3 x-6)+1=7$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 3\).

Step 1: Distribute and Simplify

First, apply the distributive property to the expression on the left side of the equation: \(2(3x - 6)\). This yields \(2 \cdot 3x - 2 \cdot 6 = 6x - 12\). So, the equation becomes: \[6x - 12 + 1 = 7\]. Now, simplify the left side by combining like terms: \(-12 + 1 = -11\), resulting in \[6x - 11 = 7\].

Step 2: Add to Both Sides

To isolate the term with \(x\), add 11 to both sides of the equation. This eliminates the \(-11\) on the left side: \[6x - 11 + 11 = 7 + 11\]. Simplify this to obtain \[6x = 18\].

Step 3: Solve for x

Now, divide both sides of the equation by 6 in order to solve for \(x\): \(\frac{6x}{6} = \frac{18}{6}\). Simplifying both sides gives \[x = 3\].

Key Concepts

Distributive Property in Linear Equations

The distributive property is an essential concept when solving linear equations. It allows us to remove parentheses and simplify expressions. Here's how it works:

You use the distributive property when you see multiplication involving a set of parentheses. For example, in the expression \(2(3x - 6)\), the 2 is multiplied by both the terms inside the parentheses. You do this by multiplying \(2\) with \(3x\) and \(-6\) separately:

• \(2 \times 3x = 6x\)
• \(2 \times -6 = -12\)

After distributing, the expression becomes \(6x - 12\). This step is crucial because it simplifies the equation, making it easier to solve as we proceed to the next steps.

Combining Like Terms

Once the distributive property has been applied, the next step in solving linear equations is to combine like terms. Like terms are terms that contain the same variable raised to the same power. For instance, consider \(6x - 12 + 1\):

• Here the constant terms are \(-12\) and \(+1\).
• Combing them by performing the operation: \(-12 + 1 = -11\).

This step reduces the equation further, which now turns into \(6x - 11\). Combining like terms simplifies the equation and helps in focusing on terms involving the variable to solve for.

Isolating Variables

Isolating variables involves rearranging the equation to get the variable you are solving for on one side of the equation. This is done by performing operations that "undo" whatever is currently being done to the variable:

In the equation \(6x - 11 = 7\), to isolate \(x\), you need to eliminate the \(-11\) from the left side. You accomplish this by adding \(11\) to both sides:

• \(6x - 11 + 11 = 7 + 11\)

This simplifies to \(6x = 18\). By doing this, we have successfully isolated \(6x\) and are one step closer to finding the value of \(x\).

Basic Algebra Steps

Basic algebra steps comprise a logical sequence of actions which lead to the solution of an equation. Once variables are isolated, we apply these basic steps:

For our equation \(6x = 18\), the next step is to solve for \(x\). Since \(x\) is multiplied by \(6\), you divide both sides by \(6\) to undo this multiplication:

• \(\frac{6x}{6} = \frac{18}{6}\)

This operation yields \(x = 3\). By following simple actions — distribute, combine, and isolate — any linear equation, no matter how complex, can ultimately be solved with these foundational algebraic principles.