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Chapter 2: Problem 44

Solve. $$ 4 x+5=5 $$

Short Answer

Expert verified
The solution is \( x = 0 \).

Step by step solution

01

Identify the Equation

The original equation given is \[ 4x + 5 = 5 \].Our task is to solve for the variable \( x \).
02

Isolate the Variable Term

To solve for \( x \), we need to get the terms involving \( x \) on one side of the equation. Start by subtracting 5 from both sides:\[ 4x + 5 - 5 = 5 - 5 \]\[ 4x = 0 \].
03

Solve for the Variable

Since \( 4x = 0 \), divide both sides by 4 to solve for \( x \):\[ \frac{4x}{4} = \frac{0}{4} \]\[ x = 0 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Variables
In algebra, isolating the variable is a critical step when solving linear equations. The main goal is to have the variable, such as \( x \), appear by itself on one side of the equation. This often allows us to find its value directly. The process begins by identifying terms that need to be moved away from the variable.
  • Remove constants: Begin by eliminating any numbers added or subtracted alongside the variable. For example, with the equation \( 4x + 5 = 5 \), subtract 5 from both sides.
  • Use reverse operations: Apply the opposite mathematical operations, such as turning addition into subtraction or vice versa, to maintain the equality of the equation.
By isolating the variable, you effectively simplify the problem to its core, making it easier to see the solution. Remember, isolating the variable is like clearing away the clutter around it.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying equations to make them easier to solve. This technique ensures that variables and constants are appropriately handled to solve equations accurately. In the equation \( 4x + 5 = 5 \), you perform algebraic manipulation by subtracting 5 to make \( 4x \) stand alone.
  • Keep balance: Whatever operation is performed on one side must be done to the other to maintain the algebraic balance.
  • Simplification: Try to reduce complex expressions to simpler forms, making calculations straightforward.
  • Inverses: Use inverse operations, like division if multiplication is present, to further isolate variables.
This process transforms an equation, often making the solution apparent and reducing potential for errors.
Equation Solving Steps
Understanding the methodical steps to solve an equation is vital for success in algebra. Let's walk through the steps used in our original problem to solidify this concept.
First, identify the equation: \( 4x + 5 = 5 \), where the task is to solve for \( x \). Begin by isolating the variable term:
  • Subtract 5 from both sides, reducing the equation to \( 4x = 0 \).

Next, solve for the variable by dividing both sides by 4:
  • Perform the division: \( \frac{4x}{4} = \frac{0}{4} \).
  • This results in \( x = 0 \).

These steps showcase the logical flow necessary for solving linear equations. They reveal how each deliberate action leads to unraveling unknown values efficiently.

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