Problem 46 - Algebra A Combined Function

Question

Solve. See Examples 1 through 7 $$ 0.9 x-4.1=0.4 $$

Step-by-Step Solution

Answer

The solution is \(x = 5\).

Step 1: Identify Like Terms

First, we need to identify the like terms in our equation. We have the equation \(0.9x - 4.1 = 0.4\). The variable term is \(0.9x\), and the constant terms are \(-4.1\) and \(0.4\).

Step 2: Isolate the Variable Term

To isolate the variable term on one side of the equation, we add \(4.1\) to both sides. This gives us \(0.9x = 0.4 + 4.1\).

Step 3: Simplify the Equation

Next, simplify the right side of the equation: \(0.9x = 4.5\). Now, the equation is ready to be solved for \(x\).

Step 4: Solve for the Variable

To solve for \(x\), divide both sides by \(0.9\) to get \(x = \frac{4.5}{0.9}\).

Step 5: Calculate the Result

Perform the division to find the value of \(x\). We get \(x = 5\).

Key Concepts

Understanding Like Terms

In linear equations, like terms refer to terms that have the same variable raised to the same power. They make it possible to combine and simplify expressions effectively. For instance, in the equation \(0.9x - 4.1 = 0.4\), we see \(0.9x\) is the only term with the variable \(x\). The numbers \(-4.1\) and \(0.4\) are constant terms, meaning they do not contain a variable. Like terms are crucial because they help us understand how to manipulate and simplify equations. Familiarizing yourself with recognizing and combining like terms will make the process of solving equations much simpler.

When you spot like terms in an equation, look for:

• The same variable
• The same power of the variable
• Numeric constants

This recognition is the first crucial step before proceeding to solve the equation.

Isolating the Variable

Once you've identified like terms, the next step in solving a linear equation is isolating the variable. Isolating the variable means getting the variable term by itself on one side of the equation. This is often done using addition or subtraction first. In our example, the equation \(0.9x - 4.1 = 0.4\) can be manipulated by adding \(4.1\) to both sides. This cancels out the \(-4.1\) on the left and increases the right side by \(4.1\), giving us \(0.9x = 4.5\).

To successfully isolate the variable, remember:

• Use operations that cancel out numbers on the same side as the variable.
• Whatever you apply to one side, do the same to the other to maintain balance.
• Keep simplifying as you go to make the equation more manageable.

Isolating the variable is essential, as it sets the stage for solving for it directly.

Simplifying the Equation

With the variable isolated, you may need to simplify the equation further, especially on the side that contains numbers or multiple terms. Simplification helps in reducing the complexity of the equation, making it easier to solve. For example, in \(0.9x = 4.5\), the right side has already been simplified to a single constant. Sometimes, simplification might involve combining more like terms or performing arithmetic operations like addition or subtraction.

While simplifying:

• Perform basic arithmetic to reduce numbers.
• Combine like terms, if necessary.
• Aim to have one term with the variable and one constant.

Simplification ensures the equation is in a form easy to handle, ready for the next step of solving it.

Solving for the Variable

The ultimate goal of any linear equation is to solve it, which means finding the value of the variable. Once your equation is simplified, as with \(0.9x = 4.5\), solving for the variable becomes straightforward. Divide both sides by the coefficient of \(x\), which is \(0.9\) in this scenario. This means performing the operation \(x = \frac{4.5}{0.9}\). Calculating this division results in \(x = 5\).

When solving for the variable, keep these tips in mind:

• Perform the inverse operation to isolate \(x\). If your equation involves multiplication, you will divide, and vice versa.
• Ensure your operations are balanced, affecting both sides equally.
• Check your solution by substituting it back into the original equation to verify it's correct.

Solving for the variable is the final step, revealing the solution to your equation with precision and clarity.