Problem 8 - Algebra A Combined Function

Question

Solve each equation. See Examples 1 and \(2 .\) $$ 3(2-5 x)+4(6 x)=12 $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{2}{3} \).

Step 1: Distribute the Constants

First, distribute the constants in front of the parentheses to both terms inside the parentheses. This will help simplify the equation. Apply the distributive property:\[ 3(2) - 3(5x) + 4(6x) = 12 \]Calculate the distributions:\[ 6 - 15x + 24x = 12 \]

Step 2: Combine Like Terms

Next, combine the like terms on the left side of the equation. This involves combining the terms with \( x \) and the constant terms:\[ 6 + (24x - 15x) = 12 \]Simplify the equation:\[ 6 + 9x = 12 \]

Step 3: Isolate the Variable Term

To isolate the term with \( x \), subtract 6 from both sides of the equation. This will move the constants to one side:\[ 6 + 9x - 6 = 12 - 6 \]This simplifies to:\[ 9x = 6 \]

Step 4: Solve for the Variable

To solve for \( x \), divide both sides of the equation by 9. This isolates \( x \):\[ \frac{9x}{9} = \frac{6}{9} \]Simplifying gives:\[ x = \frac{2}{3} \]

Key Concepts

Distributive Property

The distributive property is a fundamental tool in algebra that allows you to simplify expressions. To "distribute" means to multiply the term outside the parentheses by each term inside the parentheses. It's like sharing equally among all parts involved.
For example, in the equation given:

• Start with the expression: \( 3(2-5x) + 4(6x) = 12 \).
• Apply the distributive property by multiplying 3 with both 2 and \(-5x\), as well as 4 with \(6x\).
• This yields: \( 6 - 15x + 24x \).

This step helps break down complex expressions into simpler components, making the equation more manageable.

Combining Like Terms

Once you've distributed, the next step is to "combine like terms." Like terms are terms that contain the same variable raised to the same power. This means they can be added or subtracted together.
In the equation:

• Identify like terms: \(-15x\) and \(24x\).
• Combine them by performing the operation \(-15x + 24x\).
• This simplifies to \(9x\).

With like terms combined, the equation becomes easier to solve: \(6 + 9x = 12\). This step reduces the complexity of the equation by focusing only on terms that are similar.

Isolating the Variable

Isolating the variable means rearranging the equation to have the variable term on one side and constants on the other. This helps focus your attention solely on the variable you are solving for.
In the context of our equation:

• The equation is \(6 + 9x = 12\).
• To isolate \(9x\), subtract 6 from both sides.
• This results in the equation \(9x = 6\).

Isolating the variable simplifies the expression even further, prepping it for the final step of solving for the variable.

Solving for the Variable

In the final step, solve for the variable by performing operations that will leave the variable alone on one side of the equation.
From the isolated equation \(9x = 6\):

• Divide both sides by 9 to isolate \(x\).
• Calculating \(\frac{6}{9}\) simplifies to \(\frac{2}{3}\).
• Thus, \(x = \frac{2}{3}\).

This results in a solution for the variable, giving a clear value for \(x\) that satisfies the original linear equation. Understanding this process can empower you to handle various algebraic equations efficiently.