Problem 37 - Prealgebra

Question

Using the addition property of equality first, solve each of the following equations. $$-3 y-2=1$$

Step-by-Step Solution

Answer

The solution is \( y = -1 \).

Step 1: Start by isolating the variable term

To start solving the equation \(-3y - 2 = 1\), we need to isolate the term with the variable \(y\). Begin by getting rid of the constant term (-2) on the left side. Do this by adding 2 to both sides of the equation: \[ -3y - 2 + 2 = 1 + 2 \]This simplifies to: \[ -3y = 3 \]

Step 2: Solve for the variable by dividing

With \(-3y = 3\), the next step is to solve for \(y\). Since \(-3y\) means \(-3\) times \(y\), divide both sides by \(-3\) to solve for \(y\):\[ y = \frac{3}{-3} \]This simplifies to:\[ y = -1 \]

Key Concepts

Understanding Solving Linear Equations

Solving linear equations involves finding the unknown value that makes the equation true. Linear equations look like a straight line if you were to graph them on a coordinate plane. They come in the form of an equation like x + 3 = 7 or 2x - 5 = 11. A solution is found by identifying the value of the variable, often symbolized by letters like x or y. In these examples, x or y represent unknown numbers we need to find. Solving these equations basically means "uncovering" what the variable stands for. There are several methods for solving linear equations. The addition property of equality is often used first. This method involves adding or subtracting numbers from both sides to simplify the equation. This property is cornerstoned on keeping the equation balanced. If you add or subtract a number on one side of the equation, you must do the same on the other side to maintain equality. In elementary algebra, focusing first on eliminating the constants or numbers without variables tends to make the process simpler. Once the variables are isolated, you can move forward in ultimately solving for the variable.

Isolating Variable Terms in Equations

Isolating the variable term is a crucial step in solving equations. This means we aim to get the variable (like x or y) alone on one side of the equation. By doing this, we can concentrate on finding its specific value. To isolate a variable, first look at the terms that are grouped with the variable itself. Take the example -3y - 2 = 1. Here, we need to isolate the -3y from the -2. Using the addition property of equality, we add 2 to both sides of the equation as shown: -3y - 2 + 2 = 1 + 2. This step effectively eliminates the -2, leaving us with -3y = 3. By isolating the variable at this stage, we simplify the equation, setting it up nicely to solve for y. The beauty of this approach is it keeps the equation balanced while moving terms around. Remember, consistency is key; whatever operation you perform on one side, do it on the other. This way, isolation of the variable sets the path to the final solution.

Navigating Elementary Algebra Fundamentals

Elementary algebra serves as the foundational stone for more advanced mathematics. It deals with symbols and numbers, and their relationships through expressions and equations. Understanding the basics of algebra opens the door to exploring complex topics with greater ease. At its core, elementary algebra encompasses the use of numbers and letters to represent problems. With equations, the goal is often to solve for unknown variables, requiring moves that keep the equation balanced throughout the process. Common techniques include addition, subtraction, multiplication, and division. These operations help us reshape equations, ultimately leading to the unknown value. In our case with -3y - 2 = 1, the elementary concept of using the addition property of equality and balancing both sides comes into play. The simplicity of elementary algebra lies in its direct operations and systematic approaches. Learning these basics enables one to tackle larger and more complex mathematical challenges with confidence. With thorough practice and focus on each operation's impact, one can quickly advance in understanding and applying these principles effectively. Developing a strong grasp of elementary algebra not only fuels your mathematical journey but enhances problem-solving skills in everyday life as well.