Problem 57 - Elementary Algebra

Question

Solve. $$ 13 x-29=1 x $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{29}{12} \).

Step 1: Isolate the Variable Terms

First, we need to get all the terms with the variable \( x \) on one side of the equation. We do this by subtracting \( 1x \) from both sides of the equation. Therefore: \[13x - 29 - 1x = 1x - 1x \]This simplifies to: \[12x - 29 = 0\]

Step 2: Eliminate the Constant Term

Next, we remove the constant term \(-29\) from the left side to begin isolating \( x \). We add \( 29 \) to both sides of the equation: \[12x - 29 + 29 = 0 + 29 \]Simplifying this, we get: \[12x = 29\]

Step 3: Solve for x

Finally, divide both sides by \( 12 \) to solve for \( x \), thus getting: \[x = \frac{29}{12}\]

Key Concepts

Isolating Variables

When solving linear equations, isolating the variable is often the key first step. This means you want to get the variable on one side of the equation by itself. In our exercise, we need to isolate the variable \( x \). To start, we make sure all \( x \) terms are on the same side. This is achieved by subtracting the smaller or additional \( x \) terms from both sides of the equation.

Imagine you have:

• \( 13x - 29 = 1x \)

To isolate \( x \), subtract \( 1x \):

• \( 13x - 1x \)

This helps combine the variable terms and makes it easier to deal with them in the following steps. Now you have \( 12x - 29 = 0 \). Isolating the variable is an essential process in simplifying equations and setting the stage for solving them.

Equation Simplification

Once the variable is isolated, simplification of the equation involves removing constants and reducing the expression to its simplest form. Simplification makes an equation more manageable and closer to the solution. In our current scenario, we focus on eliminating constants from the side with the variable.

For example, take the equation from the isolated step:

• \( 12x - 29 = 0 \)

To simplify further, we need to remove \(-29\) by adding \( 29 \) to both sides of the equation:

• \( 12x - 29 + 29 = 0 + 29 \)

Simplifying gives us \( 12x = 29 \). With the constant removed, what remains is a straightforward equation with a single term including the variable, ready to be solved.

Algebraic Operations

Algebraic operations are the steps we take to solve equations, like addition, subtraction, multiplication, or division. Each operation can help transform equations and bring us closer to the solution. From our exercise, we employ these operations strategically.Firstly, we started with subtraction of \( 1x \) from both sides so:

• Removing a term: Ensures all variables are on one side.

After simplifying and isolation, we perform division, which balances the equation by making the coefficient of \( x \) equal 1. Starting from our simplified equation:

• \( 12x = 29 \)

We divide each side by \( 12 \) to solve for \( x \):

• \( x = \frac{29}{12} \)

Each operation serves a purpose, and when applied correctly, they elegantly lead us to finding the variable's value. Mastery of these steps is crucial for solving any kind of linear equation efficiently.