Problem 7 - Prealgebra

Question

Solve the equation. \(0.35 x-63.58=55.14\)

Step-by-Step Solution

Answer

The solution is \( x = 339.2 \).

Step 1: Add 63.58 to Both Sides

First, let's move the constant term on the left side of the equation to the right side. We do this by adding 63.58 to both sides of the equation. The original equation is:\[ 0.35x - 63.58 = 55.14 \]Add 63.58 to both sides:\[ 0.35x - 63.58 + 63.58 = 55.14 + 63.58 \] This simplifies to:\[ 0.35x = 118.72 \]

Step 2: Divide Both Sides by 0.35

Next, we need to isolate the variable \( x \) by dividing both sides of the equation by 0.35, the coefficient of \( x \).The equation is now:\[ 0.35x = 118.72 \]Divide both sides by 0.35:\[ x = \frac{118.72}{0.35} \]Calculate the division:\[ x = 339.2 \]

Key Concepts

Variable Isolation

Variable isolation is a crucial step in solving linear equations because it focuses on getting the variable by itself on one side of the equation. In our equation, the variable we want to isolate is \(x\). This means we need to remove any numbers or coefficients attached to \(x\).

For the equation \(0.35x - 63.58 = 55.14\), the variable isolation starts by eliminating the constant term on the left side, \(-63.58\). We do this by using the opposite operation—adding 63.58 to both sides, which balances the equation. After simplification, you end up with \(0.35x = 118.72\).

Once the constant is removed, the next step is to deal with the coefficient (which we'll explore more in the next section) to isolate \(x\) completely.

Coefficients

Coefficients are numbers that are multiplied by the variable in an equation. Understanding how to manipulate coefficients is essential for isolating the variable. In our example equation, the coefficient of \(x\) is 0.35. This means 0.35 times \(x\).

When we want to get \(x\) alone, we need to eliminate this coefficient. After removing the constant term, we have the equation \(0.35x = 118.72\).

To isolate \(x\), we perform an operation that "undoes" the multiplication—this means dividing both sides of the equation by the coefficient, which is 0.35 in this case. This gives us \(x = \frac{118.72}{0.35}\). Subsequently, calculating the division results in \(x = 339.2\). This process of handling coefficients helps in finding the value of the variable.

Basic Arithmetic Operations

Basic arithmetic operations include addition, subtraction, multiplication, and division. These operations are the building blocks of algebra that we use to manipulate equations. When solving linear equations, applying these operations strategically is key.

In the equation \(0.35x - 63.58 = 55.14\), we see usage of several basic operations:

• Addition: We start by adding 63.58 to both sides to eliminate the negative constant.
• Division: After getting \(0.35x = 118.72\), we divide through by 0.35 to solve for \(x\).

By applying these operations correctly, we maintain the equality and steadily work towards isolating \(x\). These seemingly simple steps form the foundation of solving linear equations.

Solving Equations

Solving equations involves a series of logical steps aimed at finding the value of the unknown variable. The main goal is to manipulate the equation such that the variable is by itself on one side.

For the equation \(0.35x - 63.58 = 55.14\), we approach solving it by moving the constant term and adjusting via arithmetic operations. The process includes:

• Eliminating constants using addition (add 63.58 to both sides).
• Handling coefficients through division (dividing both sides by 0.35).

These actions reveal the solution, \(x = 339.2\), by transforming the equation at each step without changing its original equivalent form. Solving linear equations, like this one, helps develop problem-solving skills and is essential in learning algebra.