Problem 25 - Elementary Algebra

Question

Solve. $$ 89-1118 x=76-12 x $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{13}{1106} \).

Step 1: Isolate the Variable Terms on One Side

First, we need to move all terms involving the variable \( x \) to one side of the equation. To do this, add \( 1118x \) to both sides of the equation.\[89 - 1118x + 1118x = 76 - 12x + 1118x\]This simplifies to:\[89 = 76 + 1106x\]

Step 2: Move Constant Terms

Next, we want to isolate the variable term \( 1106x \) by moving the constant \( 76 \) to the left side. Subtract \( 76 \) from both sides of the equation.\[89 - 76 = 1106x\]Simplifying gives:\[13 = 1106x\]

Step 3: Solve for x

Now, solve for \( x \) by dividing both sides by \( 1106 \).\[x = \frac{13}{1106}\]Simplifying the fraction shows the solution.

Key Concepts

Isolating the Variable

In solving linear equations, one key step is isolating the variable. This means we aim to have the variable, often represented as \( x \), alone on one side of the equation. To achieve this, we need to move all the terms with the variable to one side. In our example equation, we initially have:

• \( 89 - 1118x = 76 - 12x \)

By adding \( 1118x \) to both sides, we effectively eliminate the term \( -1118x \) on the left side. It is crucial because simplifying makes it easier to handle the equation step-by-step. The equation becomes:

• \( 89 = 76 + 1106x \)

After the variable terms are isolated, focusing on simplifying and solving becomes more straightforward. This process makes equations easier to manage, leading us closer to the solution.

Solve for X

Once the variable \( x \) is isolated, the next target is to solve for \( x \). Solving means to express \( x \) explicitly and find its numerical value. After isolating \( x \), our example equation looked like:

• \( 89 = 76 + 1106x \)

The goal here is to remove any constants from the side where \( x \) is located. We achieve this by subtracting \( 76 \) from both sides, which simplifies the equation to:

• \( 13 = 1106x \)

Finally, to solve for \( x \), you need to divide both sides by \( 1106 \). This operation leads to:

• \( x = \frac{13}{1106} \)

Doing this gives us the precise value of \( x \), completing our journey to solve the equation.

Simplify the Equation

Simplification often goes hand-in-hand with solving equations. It involves reducing the equation to its most basic form, making it easier to interpret and solve. An important part of simplification involves performing operations like addition, subtraction, or division in a way that reduces complexity.In the given problem, simplifying begins when we combine like terms. Initially, we tackled:

• \( 89 - 76 = 1106x \)

Performing the subtraction on the left side, we arrived at:

• \( 13 = 1106x \)

This process is crucial because further simplifies the equation and unveils itself as a number rather than a variable or a combination of terms. As we divide to solve for \( x \), simplifying the potential fraction \( \frac{13}{1106} \) is the last step. Simplification helps avoid errors, especially when dealing with large numbers or multiple steps, and can often make final answers easier to interpret.