Problem 37 - Elementary Algebra

Question

Simplify. $$ 10 y-30-15 y $$

Step-by-Step Solution

Answer

The simplified expression is \(-5y - 30\).

Step 1: Identify Like Terms

Look at the expression: \(10y - 30 - 15y\). We need to identify and group like terms, which means identifying terms that have the same variable raised to the same power. Here, \(10y\) and \(-15y\) are like terms because they both contain the variable \(y\).

Step 2: Combine Like Terms

To simplify, combine the like terms \(10y\) and \(-15y\). This means adding the coefficients of these terms. Calculate the sum: \(10 - 15 = -5\). Therefore, \(10y - 15y = -5y\).

Step 3: Write the Simplified Expression

Now that the like terms are combined, the expression becomes \(-5y - 30\). This is the simplified form of the original expression.

Key Concepts

Like Terms

In algebra, terms that have the same variables raised to identical powers are known as 'like terms.' It's crucial to quickly spot like terms when simplifying an expression. For example, in the expression \(10y - 30 - 15y\), both \(10y\) and \(-15y\) have the variable \(y\) raised to the power of one. They are like terms because they share this commonality of variable and power. A constant, like \(-30\), does not have a variable, so it is not considered a like term with those containing \(y\). You simply look for matching variables to group them, which is the first step in simplification.

Combining Coefficients

Once like terms are identified, the next step involves combining these terms to simplify the expression. This is done by adding or subtracting the coefficients - the numerical part - of these terms. In our example, the coefficients of \(10y\) and \(-15y\) are 10 and -15, respectively.

Here’s how you do it:

• Add the coefficients: \(10 + (-15)\).
• The result is \(-5\), so the combined term is \(-5y\).

Combining coefficients helps to reduce the number of terms in an expression, moving you closer to the simplest form.

Simplified Expression

Finally, once all like terms are combined, you arrive at a 'simplified expression.' This is an expression where no further like terms can be combined. It is the most concise and manageable version of the original equation. From the expression \(10y - 30 - 15y\), after combining the terms \(10y\) and \(-15y\), the result is \(-5y - 30\).

• There are no more like terms to combine.
• The expression is now in its simplest form.

Simplified expressions are easier to work with, especially when used in larger equations or functions. They reduce complexity and potential for errors in further calculations.