Problem 37 - Algebra A Combined Function

Question

Solve. $$ 13 x-3=14 x $$

Step-by-Step Solution

Answer

x = -3

Step 1: Set the equation

We begin with the given equation, which is:\[ 13x - 3 = 14x \]

Step 2: Isolate the terms

Subtract \( 13x \) from both sides of the equation to isolate the variable terms on one side:\[ 13x - 3 - 13x = 14x - 13x \]

Step 3: Simplify the equation

Simplify both sides to get:\[ -3 = x \]

Step 4: Solve for x

The equation simplifies directly to \( x = -3 \). This is the solution to the equation.

Key Concepts

Isolating Variables

When solving linear equations like \(13x - 3 = 14x\), one of the key steps is isolating variables. The idea is to get all the terms with the variable \(x\) on one side of the equation and the constant terms on the other side. This process makes it easier to solve for the variable.
The given equation has \(x\) terms on both sides. To isolate the variable, you must eliminate the \(x\) term from one side. A common technique is to subtract one of the variable terms from both sides.

• For the equation \(13x - 3 = 14x\), subtract \(13x\) from both sides to isolate the \(x\) terms.
• This results in the equation \(-3 = x\).

By following this method, you can effectively maneuver through equations and bring the focus clearly on the isolated variable, making the solution straightforward.

Simplifying Equations

Once you've isolated the variable terms, the next important step is simplifying the equation. Simplifying is about reducing the equation to its simplest form so that you can clearly see and solve the variable. It's like tidying up a room to spot exactly what you're looking for!
After isolating the \(x\) term by subtracting \(13x\), the equation becomes \(-3 = x\). This is already a simplified form. You should aim to reduce the equation to such a state, where it's evident what value the variable holds.

• Combine like terms wherever possible.
• Aim for the variable to be on one side by itself.
• Make sure constants are on the opposite side.

Simplification removes any complexity and highlights the solution in a clear, concise manner.

Basic Algebra Concepts

Understanding the basic concepts of algebra is crucial when solving equations. Algebra involves the manipulation of variables through various operations, like addition, subtraction, multiplication, and division. Grasping these foundational ideas will make solving any equation a smooth process.
Some essential algebraic concepts you should know:

• Variables represent unknown numbers, like our \(x\) in the example.
• Operations with variables follow the same rules as regular numbers.
• Balanced operations: whatever you do to one side of the equation, do the same to the other side.

Grasping basic algebra helps you develop methods for tackling more complex problems. It's all about understanding the rules of balancing and working systematically to find solutions.