Problem 81 - Elementary Algebra

Question

Solve. $$ 52 x 2-203 x=0 $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{104}{203}\).

Step 1: Understanding the Equation

The given equation is \( 52 \times 2 - 203x = 0\). This is a linear equation with one unknown variable, \(x\). Our goal is to find the value of \(x\) that makes the equation true.

Step 2: Simplify the Equation

First, simplify the left side of the equation. Calculate \(52 \times 2\) to get \(104\). The equation becomes \(104 - 203x = 0\).

Step 3: Rearrange to Isolate the Variable

To isolate \(x\), we need to rearrange the equation. Add \(203x\) to both sides to get \(104 = 203x\).

Step 4: Solve for the Variable

Divide both sides by \(203\) to solve for \(x\), so the equation becomes \(x = \frac{104}{203}\). This fraction is in its simplest form.

Key Concepts

Solving Linear Equations

Linear equations are one of the foundational concepts in algebra, often appearing as straight lines when graphed. To solve a linear equation means to find the value of the variable that makes the equation true. In our case, we started with the equation: \[ 52 \times 2 - 203x = 0 \]. The key steps involve manipulating the equation using operations such as addition, subtraction, multiplication, and division while maintaining equality.
To solve a linear equation:

• Identify and understand the equation format.
• Simplify expressions if needed.
• Perform operations to isolate the variable and solve for its value.

By carefully following these steps, we can solve linear equations accurately.

Isolating Variables

Isolating the variable is a crucial step in solving linear equations. It involves rearranging the equation to get the unknown variable on one side and all constants on the opposite side. In our exercise, after simplifying the equation to \[ 104 - 203x = 0 \], our aim was to isolate \( x \).
To do this, we added \( 203x \) to both sides, which resulted in \[ 104 = 203x \].This strategic addition effectively moved all terms containing \( x \) to one side, making it clearer and simpler to solve.
When isolating variables, remember:

• Keep the balance by performing the same operation on both sides.
• Look for terms that need to be moved to isolate the variable efficiently.
• Use the inverse operations (such as adding to counteract subtraction) wisely.

This technique is indispensable for solving not only linear but also more complex equations.

Simplifying Equations

Simplifying is the process of condensing an equation, making it less congested and easier to handle. In the provided example, we started by calculating \( 52 \times 2 \) to simplify our equation to \[ 104 - 203x = 0 \].
Simplification can include:

• Performing arithmetic operations.
• Combining like terms.
• Simplifying fractions or radicals.

Using simplification helps in:

• Reducing complexity in subsequent steps.
• Clearly highlighting the variable of interest.
• Ensuring accuracy and efficiency in problem-solving.

Simplifying early on can make the process of solving equations much more straightforward.