Problem 53 - Prealgebra

Question

Solve each equation. $$2 x+3=9$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 3\).

Step 1: Understanding the Equation

The equation given is \(2x + 3 = 9\). Our goal is to isolate \(x\) on one side of the equation to solve for its value.

Step 2: Subtracting from Both Sides

To begin isolating \(x\), subtract 3 from both sides of the equation to eliminate the constant term next to \(x\). The equation becomes: \(2x + 3 - 3 = 9 - 3\), simplifying to \(2x = 6\).

Step 3: Dividing Both Sides

Now, divide both sides of the equation by 2 to solve for \(x\). The equation is \(\frac{2x}{2} = \frac{6}{2}\), simplifying to \(x = 3\).

Key Concepts

Basic Algebra

Algebra is the branch of mathematics that uses symbols, often letters, to represent numbers and quantities in equations and formulas. In basic algebra, these symbols, usually known as variables, represent unknown values that we solve for.
In the exercise given, the challenge is to find the value of the variable "\(x\)." This is a common task in algebra when dealing with simple linear equations. Variables like "\(x\)" in algebra are placeholders for numbers that we need to figure out.
Basic algebra involves operations like addition, subtraction, multiplication, and division, all of which are used to manipulate and solve equations. Understanding these operations and how they interact with each other allows us to solve equations and uncover the values of unknowns.
Here's a quick reminder of some key algebraic concepts:

• Variables: Symbols that represent unknown numbers.
• Constants: Known numbers in the equation.
• Operators: Symbols that indicate the operations to perform on the numbers, like \(+\), \(-\), and \(\times\).
• Terms: The parts of the equation that are separated by + or - symbols.

Equation Solving Steps

Solving linear equations is an essential skill in algebra, as it forms the foundation for more complex mathematical concepts. The process involves systematic steps to ensure accuracy. In our problem, the equation is \(2x + 3 = 9\). Let's take a closer look at how each step works.
First, identify and understand the equation. Our goal is to isolate the variable, which means getting \(x\) by itself on one side of the equation:

• **Step 1: Subtract to Remove Constants**
  Start by removing the constant on the side with the variable. Subtract 3 from both sides to maintain the balance of the equation: \(2x + 3 - 3 = 9 - 3\). This simplifies to \(2x = 6\).
• **Step 2: Divide to Isolate the Variable**
  Divide both sides of the equation by the coefficient of \(x\), which is 2. This step isolates \(x\): \(\frac{2x}{2} = \frac{6}{2}\). Simplifying further, we find that \(x = 3\).

Remember, each step must treat both sides of the equation equally to keep it balanced.

Prealgebra Concepts

Before diving deep into algebra, prealgebra provides the foundational concepts needed to tackle more advanced problems. Prealgebra typically involves understanding basic mathematical operations, properties, and concepts that prepare students for algebra.
In our problem, several prealgebra concepts come into play:

• **Understanding Operations**: Addition and subtraction are used to balance the equation. These operations are usually learned first in prealgebra.
• **Properties of Equality**: When solving equations, the principle that if you do something to one side of an equation, you must do it to the other is crucial. This maintains the balance and is a core idea in both prealgebra and algebra.
• **Number Sense**: Developing an intuitive understanding of numbers and their relationships helps when manipulating equations, like knowing that subtracting 3 from both sides maintains equality in \(2x + 3 = 9\).

Grasping these prealgebra concepts allows students to build confidence and skill in solving more complex algebraic equations.