Problem 16 - Prealgebra

Question

Solve each equation. $$a+3=-1$$

Step-by-Step Solution

Answer

The solution is \( a = -4 \).

Step 1: Identify the Equation

The given equation is \( a + 3 = -1 \). Our goal is to solve for \( a \).

Step 2: Isolate the Variable

To isolate \( a \) on one side of the equation, we need to eliminate the constant term. We do this by subtracting 3 from both sides of the equation: \( a + 3 - 3 = -1 - 3 \).

Step 3: Simplify Both Sides

Simplifying the left side, \( a + 0 \) simplifies to \( a \). Simplifying the right side, \( -1 - 3 \) gives \( -4 \).

Step 4: Write the Solution

The solution to the equation is \( a = -4 \).

Key Concepts

Solving Equations

Solving equations in algebra involves finding the unknown variable which balances both sides of the equation. It's like solving a mystery: you seek to find the missing piece that completes the puzzle. When given an equation such as \( a + 3 = -1 \), your task is to determine the value of \( a \).
This means taking steps to manipulate the equation so you can isolate the variable, \( a \), and determine its value. It involves operations, such as addition, subtraction, multiplication, or division, to both sides to keep the equation balanced. Think of it as a seesaw—whatever you do to one side, you must do to the other in order to keep balance.
Solving simple algebraic equations prepares you for more complex problems and helps develop logical thinking and problem-solving skills.

Isolating Variables

Isolating the variable is a key step in solving equations. It refers to manipulating an equation so that the variable you need to solve for stands alone on one side of the equation. In the equation \( a + 3 = -1 \), isolating the variable means getting \( a \) by itself.
Here's how you do it:

• Look at what is being added, subtracted, multiplied, or divided onto the variable.
• Use the inverse operation to remove the extra terms. In the case of \( a + 3 = -1 \), you subtract 3 from both sides because the inverse of addition is subtraction.
• This leaves you with \( a = -1 - 3 \), which simplifies to \( a = -4 \).

By isolating \( a \), you successfully find the value that makes the equation true. This process is fundamental in algebra as it sets the stage for handling more intricate equations.

Basic Algebra

Basic algebra forms the foundation of mathematics beyond simple arithmetic. It introduces variables as placeholders for numbers and uses symbols to express mathematical ideas and relationships. In algebra, equations like \( a + 3 = -1 \) represent problems needing solutions to unknown values.
Understanding algebra is essential because:

• It introduces the concept of variables, which are crucial for solving real-world problems.
• It teaches how to manipulate equations, which is useful in fields ranging from science to finance.
• It develops critical thinking skills, encouraging you to approach problems logically and systematically.

When you learn basic algebra, it enhances your mathematical literacy and prepares you for advanced studies. It reveals how different variables can interact and affect each other, showing the interconnectedness of numbers and operations in dynamic ways.