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Chapter 2: Problem 109

Solve: \(2 x-3=5\)

Short Answer

Expert verified
The solution is x = 4.

Step by step solution

01

Isolate the variable term

To start solving the equation, we first want to move the constant term to the other side. We do this by adding 3 to both sides.\[ 2x - 3 + 3 = 5 + 3 \]This simplifies to:\[ 2x = 8 \]
02

Solve for the variable

Now that the equation is simplified to \(2x = 8\), the next step is to solve for \(x\) by dividing both sides of the equation by 2.\[ \frac{2x}{2} = \frac{8}{2} \]This gives us:\[ x = 4 \]

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Key Concepts

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

Solving Equations
When we talk about solving equations, we are referring to finding the value of the variable that makes the equation true. In linear equations like the one given, these typically involve one variable and can be solved through basic arithmetic operations.
The main goal when solving any equation is to isolate the variable on one side of the equation, making it equal to a constant on the other side. We achieve this by performing operations that keep the equation balanced on both sides.
  • Identify the operations applied to the variable.
  • Use inverse operations, like addition if the variable is subtracted from, to eliminate constants attached to the variable.
  • Perform the same operations on both sides to maintain equality.
Solving equations involves understanding and applying these principles systematically to reveal the variable's value.
Isolation of Variables
Isolation of variables is a crucial step in solving equations. It refers to the process of rearranging an equation so that one side contains only the variable we are solving for. For example, in the equation \(2x - 3 = 5\), we aim to have the variable \(x\) alone on one side of the equation.
To isolate the variable, consider the equation as a balanced scale. Any operation on one side must be done on the other to keep it balanced.
  • Look at the term involving the variable and the constant terms.
  • Move the constants away from the variable by using opposite operations (e.g., adding the opposite).
  • The goal is to end with the variable by itself on one side of the equation.
This technique is essential for simplifying the equation, thereby making it easier to solve for the variable.
Algebraic Manipulation
Algebraic manipulation is the series of operations applied to equation terms to simplify or rearrange them. It's a cornerstone for solving equations because it enables us to transform the equation into a more workable form. In our solved example, algebraic manipulation involves both addition and division.
First, we used addition to move constants that were subtracted from the variable term. This simplifies the equation by reducing it to fewer terms.
  • Addition and subtraction are used to clear constants.
  • Multiplication and division adjust coefficients of variables.
  • These operations are fundamental for maintaining equivalence in transformations.
By carefully planning which algebraic operations to apply, we maintain the balance and equivalency needed to solve for the unknown variable effectively.

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