91Ó°ÊÓ

When solving a linear equation, one of the primary goals is to get the variable by itself on one side of the equation. This is what 'isolating the variable' means. In the given exercise, we start with the equation:

• \( y - 15.9 = -3.8 \)

To isolate \( y \), we need to eliminate the \( -15.9 \) from the left side of the equation. We do this by adding 15.9 to both sides to maintain the equality. The equation then turns into:

• \( y - 15.9 + 15.9 = -3.8 + 15.9 \)

As a result, the \( -15.9 \) and \( +15.9 \) cancel each other out on the left, leaving us with the simplified form:

• \( y = 12.1 \)

Adding the same value to both sides ensures the balance of the equation stays intact. The concept of isolating the variable is crucial across many types of mathematical problems, enabling us to clearly see the value of the unknown.

Understanding like terms is an integral part of algebra, helping to simplify expressions efficiently. Like terms are terms that have the same variable raised to the same power. Their coefficients may be different, but their variables or lack thereof must match to be combined.
In the exercise given, we don't directly work with combining like terms. However, ensuring we understand them is important for more complex scenarios. Typically, like terms can be:

• Variable terms, like \( 3x \) and \( 7x \)
• Constant terms, like \( -15.9 \) and \( +15.9 \)

We are combining constant terms in this problem by adding 15.9 to both the left and right side. Understanding that these constants can combine to zero on one side simplifies the isolation of our variable \( y \). If we encounter like terms in other equations, we'll want to be sure they are combined before performing further operations to isolate variables.

Simplifying expressions is about making an equation or expression as concise and clear as possible. This usually involves combining like terms, removing parentheses, and performing arithmetic operations.
In our example problem, the simplification happens when we remove the terms that cancel each other out:

• \( y - 15.9 + 15.9 \) simplifies to \( y \)
• \( -3.8 + 15.9 \) simplifies to \( 12.1 \)

By adding 15.9 to both sides, we completed a straightforward arithmetic operation that simplified both sides of the equation to reach

• \( y = 12.1 \)

In more complex equations, simplifying might mean combining like terms and applying distributive properties or exponent rules. Ultimately, simplifying reduces complexity and gives a clearer picture of what the equation represents.