Problem 13 - Elementary Algebra

Question

Solve. $$ 9 x-25=12 x-25 $$

Step-by-Step Solution

Answer

The solution is \(x = 0\).

Step 1: Simplify the Equation

Start by simplifying both sides of the equation to make it easier to solve. The equation is already simplified in its basic form: \(9x - 25 = 12x - 25\).

Step 2: Eliminate the Constant Terms

To eliminate the constant \(-25\) on both sides, add \(25\) to each side of the equation:\[9x - 25 + 25 = 12x - 25 + 25\]This simplifies to:\[9x = 12x\].

Step 3: Solve for the Variable

Now, we need to get all the \(x\) terms on one side. Subtract \(9x\) from both sides to isolate \(x\):\[9x - 9x = 12x - 9x\]This further simplifies to:\[0 = 3x\].

Step 4: Find the Value of the Variable

To solve for \(x\), divide both sides by \(3\):\[\frac{0}{3} = \frac{3x}{3}\]This results in:\[0 = x\].

Key Concepts

Solving Linear Equations

Linear equations are equations where the highest power of the variable is one. These equations take the form \( ax + b = cx + d \), where \( x \) is the variable and \( a, b, c, \) and \( d \) are constants.

• The goal is to determine the value of \( x \), which satisfies the equation.
• Linear equations are found everywhere, from calculating distances to balancing a budget.

The key to solving them is to manipulate the equation using operations such as addition, subtraction, multiplication, or division, to eventually isolate the variable and find its value. It involves combining like terms and applying inverse operations. A systematic approach ensures accuracy and efficiency.

Isolating Variables

The process of isolating variables involves rearranging the equation so that the variable is by itself on one side of the equation. Let's walk through how this works. First, you look to eliminate constants or coefficients that are crowding the variable:

• You can do this by adding or subtracting numbers on both sides of the equation.
• This balances the equation while stripping away numbers away from the variable.

Once the constant terms are dealt with, focus on the coefficients attached to the variable. This can be done by performing operations that nullify these coefficients like:

• Dividing both sides by the coefficient of the variable thoroughly isolates the variable.
• The end result is an expression where the variable stands alone, signifying it has been successfully isolated.

With the variable isolated, you can directly write the value that satisfies the given equation.

Simplifying Equations

Simplifying an equation is a crucial part of solving it—it makes the equation easier to handle and understand. In our original example, the equation was \( 9x - 25 = 12x - 25 \). Here's why simplification works wonders:

• First, check if there are like terms on each side of the equation. In our example, \(-25\) was on both sides, so adding \(25\) simplified the equation.
• Next, perform operations that can make the equation, hence, involving elimination of constants or simplifying complex expressions.

Why is simplification important?

• It reduces the complexity, making the solving process more straightforward.
• It highlights the core components of the equation that need focus when solving, like the terms involving the variable.

Always strive to simplify as a first step, as it sets the stage for accurately solving for the variable and verifying your solution.