Problem 49 - Prealgebra

Question

Simplify each side of the following equations first, then solve. $$5-18=3 y-2 y+1$$

Step-by-Step Solution

Answer

\( y = -14 \)

Step 1: Simplify Left Side

The left side of the equation is \( 5 - 18 \). Simplify this expression: \( 5 - 18 = -13 \). Thus, the left side simplifies to \(-13\).

Step 2: Simplify Right Side

The right side of the equation is \( 3y - 2y + 1 \). Combine like terms \( 3y - 2y \) to get \( y \). So the right side simplifies to \( y + 1 \).

Step 3: Set Simplified Sides Equal

Now, set the simplified sides equal to each other: \(-13 = y + 1\).

Step 4: Isolate \( y \)

To solve for \( y \), subtract 1 from both sides of the equation: \(-13 - 1 = y + 1 - 1\). This simplifies to \(-14 = y\).

Step 5: Conclusion

We have isolated \( y \) to find that \( y = -14 \). This is the solution to the equation.

Key Concepts

Simplifying Expressions

Simplifying expressions is the process of making complicated expressions easier to handle and understand. In the exercise, you are given an equation with terms on both sides. To simplify, you need to perform basic arithmetic operations like addition and subtraction. For instance:

• The original expression on the left side is \( 5 - 18 \), which simplifies to \( -13 \) by subtracting 18 from 5.
• Perform similar simplifications on the right side, such as combining terms like \( 3y - 2y \) to get \( y \).

Simplifying helps to reveal the simpler form of the equation, allowing you to see the core elements you need to work with. When expressions are simplified, solving becomes a much clearer process.

Combining Like Terms

Combining like terms is a key step in simplifying algebraic expressions, especially in equations. Like terms are those that have the same variables raised to the same power. In our exercise:

• On the right side of the equation, you have \( 3y - 2y + 1 \).
• Here, the terms \( 3y \) and \( -2y \) are like terms because they both contain the variable \( y \).
• You combine these terms by performing the subtraction \( 3y - 2y \) to get \( y \).

This process reduces the complexity of equations and is crucial for simplifying expressions which in turn makes solving them more straightforward.

Isolating Variables

Isolating a variable means getting the variable on one side of the equation by itself. This is crucial to solve for the variable's value. In our example, you need to isolate \( y \) from the equation \(-13 = y + 1\). Here’s how you can do it:

• Subtract 1 from both sides: \(-13 - 1 = y + 1 - 1\)
• After this operation, you end up with \(-14 = y\)

By isolating \( y \), you have deduced the solution to the equation. This step can involve addition, subtraction, multiplication, or division, depending on the equation. It's a critical skill for solving linear equations and is widely used in algebra.

Linear Equations

Linear equations are equations where the variable is raised only to the first power and does not appear in any denominators, exponents, or under a radical. They generally take the form \( ax + b = c \).In our exercise, the equation \(-13 = y + 1 \) is a linear equation. The key steps to solving it involve:

• First, simplify both sides by combining like terms and performing arithmetic operations.
• Next, isolate the variable to find its value.
• This results in a solution where \( y \) equals a number, in our case, \( y = -14 \).

Solving linear equations often lays the foundation for more complex mathematics, and recognizing their structure helps in approaching them efficiently. As you become more familiar with these forms, solving them will become a more intuitive process.