Problem 17 - Prealgebra

Question

In Exercises 17-34, solve the equation. \(6.24 x-5.2=5.2 x\)

Step-by-Step Solution

Answer

The solution is \(x = 5\).

Step 1: Move Variable Terms to One Side

First, we need to get all the terms with the variable \(x\) on one side of the equation. To do this, subtract \(5.2x\) from both sides of the equation. This will help isolate the variable. The equation will become:\[6.24x - 5.2x - 5.2 = 0\].

Step 2: Simplify the Equation

Now, let’s simplify the expression by combining like terms. Subtract \(5.2x\) from \(6.24x\) to simplify the left side:\[(6.24 - 5.2)x - 5.2 = 0\].Calculate the subtraction: \[1.04x - 5.2 = 0\].

Step 3: Isolate the Variable

Next, we want to get the \(x\) term by itself. Add \(5.2\) to both sides of the equation to get rid of the constant term on the left:\[1.04x = 5.2\].

Step 4: Solve for the Variable

Finally, divide both sides by \(1.04\) to solve for \(x\):\[x = \frac{5.2}{1.04}\].Upon division, the result is:\[x = 5\].

Key Concepts

Solving Equations

The process of solving equations is all about finding the value of the unknown variable that makes the equation true. Usually, this involves several steps to simplify the equation by using arithmetic operations—addition, subtraction, multiplication, and division.
The aim is to rearrange the equation gradually until you isolate the variable and solve for its value.

• First, identify the terms in the equation. For example, in our exercise, we have terms involving variable has coefficient as well as constant terms.
• Understand the operations involved: look out for signs and group similar operations together.
• Use inverse operations strategically to slowly unwind the equation, which will give you a clearer path toward the solution.

By practicing these steps, you can tackle almost any linear equation that comes your way. Be patient and proceed methodically!

Variable Isolation

Variable isolation is all about getting the variable you’re solving for alone on one side of the equation. This often involves moving other terms to the opposite side.
In our example, we started by subtracting the term containing the same variable on the opposite side:

• Note that each operation must be performed on both sides of the equation to keep it balanced.
• For example, subtracting the variable term like \(5.2x\) from both sides helps move all variable terms together on one side.
• Use arithmetic operations to isolate the variable.

This forms the basis for solving equations, allowing you to see clearly what value the variable should take. Once the variable is isolated, the next steps to solve the equation become more straightforward.

Combining Like Terms

Combining like terms is a crucial step in simplifying equations and making them easier to solve. Like terms are terms that have exactly the same variable parts and can be combined by adding or subtracting the coefficients.

• Look for terms with the same variable. In our given equation, these were the terms \(6.24x\) and \(5.2x\).
• Simply add or subtract the coefficients. Here, we subtracted \(5.2\) from \(6.24\) to get \(1.04x\).
• This creates a simpler expression that’s easier to work with.

It helps reduce the complexity of the equation, paving the way for more straightforward operations and a quicker path to the solution.

Equation Simplification

Equation simplification is the process of making an equation easier to work with. After combining like terms, you often need to simplify further to isolate the variable.
This involves removing constants and performing arithmetic operations until you get the simplest form of the equation.

• Start by removing constant terms on the same side as the variable by using addition or subtraction.
• In our example, we add \(5.2\) to both sides to remove the constant term next to \(x\).
• Finally, solve the equation by performing division or multiplication if needed to completely isolate the variable.

This step-by-step simplification makes it much easier to solve the equation accurately. It’s about making each subsequent step as straightforward as possible, guiding you toward the solution of \(x = 5\) in our example.