Problem 45 - Elementary Algebra

Question

Choose the appropriate method to solve the following. $$ 6(x+1) 2=11 x+7 $$

Step-by-Step Solution

Answer

The solution is \(x = -5\).

Step 1: Expand the Left Side

First, we need to simplify the left side of the equation by expanding it. Take note that you have to multiply the 6 by the expression \((x + 1)\).\[6(x + 1) = 6x + 6\]

Step 2: Simplify Left Side Further

Now that we have expanded the term, let's multiply by 2 as indicated by the equation:\[2(6x + 6) = 12x + 12\]Now the equation is \(12x + 12 = 11x + 7\).

Step 3: Bring All Terms to One Side

To solve for \(x\), we need all \(x\) terms on one side and numbers on the other. Subtract \(11x\) from both sides:\[12x - 11x + 12 = 7\]This simplifies to:\[x + 12 = 7\]

Step 4: Solve for \(x\)

To isolate \(x\), subtract 12 from both sides of the equation:\[x + 12 - 12 = 7 - 12\]This simplifies to:\[x = -5\]

Step 5: Verify the Solution

Substitute \(x = -5\) back into the original equation to ensure it holds true:\[6((-5) + 1) \, 2 =? 11(-5) + 7\]Calculate each side:\[6(-4) \, 2 = -48\quad\text{and}\quad -55 + 7 = -48\]Both sides are equal, thus confirming that \(x = -5\) is correct.

Key Concepts

Simplification

In solving linear equations, simplification is a key step. It involves making an equation easier to work with by reducing it to its simplest form. This process often includes distributing numbers across parentheses, combining like terms, and removing any unnecessary components.
For example, in the given problem, we start by handling the expression on the left. Here, we expand \( 6(x + 1) \) to \( 6x + 6 \).

• The initial step helps in working with a simpler expression.
• It makes operations like adding or subtracting terms more straightforward.
• After expanding, multiplying by 2 leads to \( 12x + 12 \), simplifying the left further.

This simplification transforms the equation, making it easier to deal with the variable terms on both sides, paving the way for a clear pathway to the solution.

Verification of Solution

After obtaining a solution, verification is crucial to ensure that the solution is indeed correct. This involves substituting the found value back into the original equation to check for consistency.
In this exercise, we found \( x = -5 \) and substituted it into the original equation:

• Left Side: \( 6((-5) + 1) \times 2 \)
• Calculate: \( 6(-4) \times 2 = -48 \)
• Right Side: \( 11(-5) + 7 \)
• Calculate: \( -55 + 7 = -48 \)

Both sides equate to -48, confirming the solution is correct. Verification helps in validating the accuracy of the solution. It reassures that no calculative errors were made in the process, and the purchased logic holds true throughout the procedure.

Expanding Expressions

Expanding expressions is a foundational skill in algebra. This step involves removing parentheses by distributing a factor across a sum or difference inside the brackets.
Consider the expression \( 6(x + 1) \). Expanding this involves:

• Applying the distributive property: multiply each term inside the parenthesis by 6.
• This results in \( 6 \times x + 6 \times 1 \), giving \( 6x + 6 \).

Understanding expression expansion:

• Prepares you for more complex manipulations.
• Ensures clarity in equations.
• Facilitates the solving of equations, as it often reveals simpler combinations or structures.

It is a strategic move that simplifies subsequent calculations, allowing us to manipulate the formula or equation effortlessly toward obtaining solutions.