Problem 30 - Algebra 1

Question

Solve the equation. $$12(2-x)=6$$

Step-by-Step Solution

Answer

The solution to this equation is \(x = 1.5\)

Step 1: Distribute Inside the Parentheses

First, distribute the 12 inside the parentheses to obtain a simplified equation: \(12 * 2 - 12 * x = 6\), which simplifies to \(24 - 12x = 6\)

Step 2: Isolate the Term With the Variable

Next, isolate the term with the variable. To do this, subtract 24 from both sides of the equation to get: \(-12x = 6 - 24\) which simplifies to \(-12x = -18\)

Step 3: Solve For the Variable

Finally, to solve for \(x\), divide both sides of the equation by -12. Thus, \(x = -18 / -12\), which simplifies to \(x = 1.5\)

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to simplify equations, making them easier to solve. It states that when you multiply a single term outside of parentheses by each term inside the parentheses, the equation remains equivalent. For example, in the equation \(12(2-x)=6\), applying the distributive property means you multiply 12 by both 2 and \(-x\).

• Multiply 12 by 2 to get 24.
• Multiply 12 by \(-x\) to get \(-12x\).

So, the equation \(12(2-x)=6\) turns into \(24 - 12x = 6\). By breaking down the equation in this manner, you eliminate parentheses and reduce complexity. This process helps set the stage for the next steps involved in solving equations.

Isolation of Variables

Isolation of variables is an important strategy used in solving equations, especially linear ones. The goal is to get the variable alone on one side of the equation. This makes finding its value straightforward. In our example, once the distributive property was applied, resulting in \(24 - 12x = 6\), the next step is to isolate \(-12x\).

• Start by subtracting 24 from both sides to move constant terms to the opposite side of the equation.
• This results in \(-12x = 6 - 24\).
• The equation simplifies to \(-12x = -18\).

With \(-12x\) isolated on one side, you can clearly see what operations need to be performed in the next step to solve for \(x\). Remember, always perform the same operation on both sides to maintain the balance of the equation.

Solving Equations

Solving equations is the process of finding the value of the variable that makes the equation true. Once you've isolated the term containing the variable, the finishing touch is solving for \(x\). From the equation \(-12x = -18\), we need to isolate \(x\). This involves dividing both sides by \(-12\) to solve for \(x\).

• Divide \(-18\) by \(-12\).
• Performing the division gives \(x = 1.5\).

It’s crucial to note that the sign changes following the division, as dividing two negative numbers yields a positive result. Solving equations requires careful calculation to ensure each step logically follows the previous one, bringing you to the solution efficiently.