Problem 2 - Prealgebra

Question

Solve the equation. \(-0.3 x-6.5 x=3.4\)

Step-by-Step Solution

Answer

x = -0.5

Step 1: Combine Like Terms

The equation is given as \(-0.3x - 6.5x = 3.4\). First, we combine the like terms on the left side of the equation. This involves adding \(-0.3x\) and \(-6.5x\) together, which gives us \(-6.8x\). So, the equation simplifies to \(-6.8x = 3.4\).

Step 2: Solve for x

To isolate \(x\), divide both sides of the equation by \(-6.8\). This setup is \(-6.8x / -6.8 = 3.4 / -6.8\). This simplifies to \(x = -0.5\).

Key Concepts

Combining Like Terms

In solving linear equations, one of the first steps often involves combining like terms. This refers to the process of merging terms in the equation that have the same variable part, such as those containing \(x\) here. When dealing with the example given, \(-0.3x - 6.5x = 3.4\), both terms on the left side are like terms because they include the variable \(x\).

To combine them, you simply add their coefficients, the numbers in front of \(x\), together:

• Start by adding \(-0.3\) and \(-6.5\).
• This results in \(-6.8x\).

Thus, the equation simplifies to \(-6.8x = 3.4\). Combining like terms helps to condense the equation, making it easier to solve by reducing the number of terms you need to work with.

Isolation of Variables

Isolating the variable is a key step in solving equations. It means getting the variable you are solving for, like \(x\) here, all by itself on one side of the equation. This process simplifies everything, allowing you to clearly see what the variable equals.

To isolate \(x\) in the simplified equation \(-6.8x = 3.4\), perform the following:

• Divide both sides of the equation by \(-6.8\) to get \(x\) alone.
• This is represented as \(-6.8x / -6.8 = 3.4 / -6.8\).

By completing this division, you simplify the equation to \(x = -0.5\). With \(x\) isolated, the solution reveals itself clearly.

Solving Equations Step by Step

Solving equations step by step is crucial for clarity and accuracy. It involves systematically addressing each part of the equation, using logical operations to simplify and solve. Let's go through the process using our example:

1. **Combine Like Terms**: Start by simplifying the equation by combining terms that share the same variable.

• In this case, \(-0.3x - 6.5x\) becomes \(-6.8x\).

2. **Isolate the Variable**: The next step involves strategically performing operations to solve for \(x\).

• Divide both sides by the coefficient of \(x\), which here is \(-6.8\).
• This isolates \(x\) and solves the equation, leading you to \(x = -0.5\).

Following these steps thoroughly ensures you correctly solve the equation while understanding each transformation. This structured approach not only helps in solving the specific equation but also builds a solid foundation for solving more complex ones. Remember, practice makes perfect, and each equation solved enhances your skills.