Problem 29 - Prealgebra

Question

Solve each equation using the methods shown in this section. $$3 x-5=11+2(x-6)$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 4\).

Step 1: Simplify both sides

Simplify both sides of the equation by distributing and combining like terms, if possible. The equation is \(3x - 5 = 11 + 2(x - 6)\). Start by expanding the right side:\[3x - 5 = 11 + 2x - 12\] Simplify the right side by combining like terms:\[3x - 5 = 2x - 1\]

Step 2: Isolate variables

To isolate the variable terms on one side of the equation, subtract \(2x\) from both sides:\[3x - 2x - 5 = -1\]This simplifies to:\[x - 5 = -1\]

Step 3: Solve for x

Now, isolate \(x\) by adding 5 to both sides of the equation:\[x - 5 + 5 = -1 + 5\]Simplify to get:\[x = 4\]

Key Concepts

Distributive Property

The distributive property is a foundational concept in algebra that allows us to multiply a single term over a sum or a difference inside parentheses. It is a handy tool when simplifying expressions. In the given exercise, the distributive property is used to expand the equation from its original form. When applying the distributive property, we multiply the outer term with each inside the parentheses.
For example, in the expression \(2(x-6)\), the number 2 is multiplied by each number inside the parentheses:

• Multiply 2 by \(x\), resulting in \(2x\).
• Multiply 2 by \(-6\), resulting in \(-12\).

After applying the distributive property, the expression becomes \(2x - 12\). This step is essential as it lays the groundwork for further simplification and solving the equation.

Combining Like Terms

Combining like terms is crucial for simplifying expressions and solving equations. Like terms are terms that have the same variable raised to the same power. By organizing these terms, we make equations smaller and easier to work with.
In our exercise, after using the distributive property, the equation has several terms on the right side: \(11 + 2x - 12\). Here, the like terms \(11\) and \(-12\) are numbers without variables. We combine them to simplify the expression:

• \(11 - 12 = -1\)
• Keep the term \(2x\) as it is, since there are no other like terms to combine it with.

Now the equation becomes \(3x - 5 = 2x - 1\). All like terms have been combined, simplifying the equation further.

Isolating Variables

Isolating the variable is the process of getting the variable alone on one side of the equation. This is the step where we work towards solving for the unknown.
In the given example, we started with the equation \(3x - 5 = 2x - 1\). The goal is to have \(x\) by itself on one side. To do this, follow these steps:

• Subtract \(2x\) from both sides to get all \(x\) terms on one side: \(3x - 2x = x\).
• This simplifies to \(x - 5 = -1\).
• Next, add 5 to both sides to cancel out \(-5\): \(x - 5 + 5 = -1 + 5\).
• Finally, this simplifies to \(x = 4\).

By isolating the variable, we solve the equation and find that \(x = 4\). Understanding this concept helps solve any linear equation with precision.