Problem 81 - Prealgebra

Question

The problems below review some of the material on solving equations. Reviewing these problems will help you with the next section. Solve. $$2(x+3)=10$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 2\).

Step 1: Expand the Equation

Start by expanding the equation. Apply the distributive property by multiplying 2 with the terms inside the parentheses: \(2(x+3) = 2 \cdot x + 2 \cdot 3\). This simplifies to \(2x + 6 = 10\).

Step 2: Isolate the Variable Term

Next, we need to get the term with the variable by itself on one side of the equation. Subtract 6 from both sides to remove the constant term on the left side: \(2x + 6 - 6 = 10 - 6\). This simplifies to \(2x = 4\).

Step 3: Solve for the Variable

Now, solve for \(x\) by dividing both sides of the equation by 2: \(\frac{2x}{2} = \frac{4}{2}\). This gives \(x = 2\).

Key Concepts

Distributive Property

When solving equations, especially linear ones, the distributive property is a crucial tool. It helps you get rid of parentheses so you can work with simpler expressions. This property lets you multiply a single term across terms inside brackets. In our example, we started with the equation \(2(x+3) = 10\). Here, the distributive property tells us to multiply each term within the parentheses, \(x+3\), by the number outside, which is 2. So you perform following steps:

• Multiply 2 by \(x\), giving \(2x\).
• Multiply 2 by 3, giving 6.

This transforms the problem into \(2x + 6 = 10\), making it easier to proceed.

Isolating Variables

The goal of solving equations is to find the value of the variable, usually represented as \(x\). To do this, you need to isolate the variable, or get it alone on one side of the equation. The first step is to eliminate any constants from the expression containing the variable.In our problem, once we reached \(2x + 6 = 10\), we tackled the constant term, which is 6. Applying subtraction, a basic arithmetic operation, we subtracted 6 from both sides:

• This cancels the 6 on the left, leaving \(2x\).
• On the right, \(10 - 6\) simplifies to 4.

Now you have a cleaner equation: \(2x = 4\). At this point, the variable term is isolated, and you can easily proceed to solve for \(x\).

Linear Equations

Linear equations are a core component of algebra and can always be represented in the form \(ax + b = c\). These equations have solutions that provide straight-line graphs when plotted. In our example, \(2x + 6 = 10\) is a simple linear equation. Solving these types of equations involves straightforward steps:

• Apply the distributive property, if necessary, to eliminate parentheses.
• Simplify each side of the equation.
• Isolate the variable using inverse operations like addition, subtraction, multiplication, or division.

In this case, after isolating the variable, we're left with \(2x = 4\). The last step for solving is dividing both sides of the equation by the coefficient of \(x\), which is 2, to find \(x = 2\). Understanding these steps allows you to solve any linear equation.