Problem 21 - Prealgebra

Question

Solve each equation using the methods shown in this section. $$4(x-6)+1=2 x-9$$

Step-by-Step Solution

Answer

The solution is \(x = 7\).

Step 1: Expand Parentheses

First, expand the parentheses in the equation. The term \(4(x-6)\) can be expanded to \(4x - 24\), so the equation becomes \(4x - 24 + 1 = 2x - 9\).

Step 2: Combine Like Terms

Combine like terms on the left side of the equation. The terms \(-24 + 1\) combine to get \(-23\), giving us the equation \(4x - 23 = 2x - 9\).

Step 3: Move Variable Terms to One Side

Move the \(2x\) on the right to the left side by subtracting \(2x\) from both sides. This results in \(4x - 2x - 23 = -9\), which simplifies to \(2x - 23 = -9\).

Step 4: Isolate the Variable Term

Add \(23\) to both sides to isolate the variable term. This gives \(2x = 14\).

Step 5: Solve for the Variable

Finally, divide both sides by \(2\) to solve for \(x\). Thus, \(x = \frac{14}{2} = 7\).

Key Concepts

Expanding Parentheses

One of the first steps when solving linear equations, like the one found in our example, is to expand any parentheses you may encounter. This action means you'll need to distribute the number or term outside the parentheses to everything inside. In our equation, we have a term of the form \(4(x-6)\).To expand this:

• Multiply the 4 with the \(x\), giving you \(4x\).
• Then, multiply the 4 with the \(-6\), which gives you \(-24\).

Putting it together results in \(4x - 24\). This step is crucial as it simplifies the expression, making it easier to work with as you proceed to solve the equation. Remember, expanding parentheses is simply about applying the distributive property of multiplication over addition or subtraction within the parentheses.

Combining Like Terms

After expanding, the equation might still look complicated. The next step is to combine like terms to simplify it further. This process involves grouping together terms that have the same variables raised to the same power.For our example equation, once expanded: \(4x - 24 + 1\), you can simplify further by combining the constants,

• \(-24\) and \(+1\) combine to become \(-23\).

This gives you the simplified equation \(4x - 23 = 2x - 9\). By combining like terms, you reduce the number of terms in the equation, making it clearer and easier to manage in subsequent steps. The key here is to look carefully for terms with identical variable parts and on free constants, and add or subtract them as necessary.

Isolating the Variable

Once like terms are combined, it’s time to isolate the variable term. The goal is to have the variable, in this case \(x\), all by itself on one side of the equation. This is achieved through a series of manipulations involving addition, subtraction, multiplication, or division.Here’s how you can isolate \(x\) in our example:

• Start by moving all terms with \(x\) to one side of the equation. We subtract \(2x\) from both sides, leading to \(4x - 2x - 23 = -9\).
• This simplifies to \(2x - 23 = -9\).
• Next, add \(23\) to both sides to remove the \(-23\) from the same side as \(x\). This results in \(2x = 14\).
• Finally, divide each side by \(2\) to solve for \(x\), giving us \(x = 7\).

Isolating the variable is often the final hurdle in solving linear equations, culminating in the solution where the variable has a specific numeric value. Remember, every move here should maintain the balance of the equation.