Problem 49 - Elementary Algebra

Question

Simplify. $$ 13 x-25 y+23 x-35 y $$

Step-by-Step Solution

Answer

The expression simplifies to \(36x - 60y\).

Step 1: Identify Like Terms

To simplify the given expression, we must first identify like terms. Like terms are terms that contain the same variable raised to the same power. In this expression, the like terms are \(13x\) and \(23x\) as well as \(-25y\) and \(-35y\).

Step 2: Group Like Terms

Next, we'll group the like terms together. The expression \(13x - 25y + 23x - 35y\) can be grouped as \((13x + 23x) + (-25y - 35y)\).

Step 3: Combine Like Terms

Now, combine the like terms from each group. For the \(x\) terms, combine \(13x + 23x\) to get \(36x\). For the \(y\) terms, combine \(-25y - 35y\) to get \(-60y\).

Step 4: Write the Simplified Expression

Finally, combine the results from Step 3 to get the simplified expression, which is \(36x - 60y\).

Key Concepts

Like Terms

In algebra, identifying like terms is key to simplifying expressions. So, what are like terms? They're terms that have exactly the same variable factors. This means the variable part must match, including the power to which they're raised. For example, in the expression \(13x - 25y + 23x - 35y\), \(13x\) and \(23x\) are like terms because they both have the variable \(x\) raised to the same power of 1. Similarly, \(-25y\) and \(-35y\) are like terms because they both contain the variable \(y\) raised to the power of 1. Identifying like terms helps in simplifying expressions by allowing us to group and combine terms efficiently. Recognizing these similarities ensures that we can manipulate expressions correctly.

Combining Like Terms

Once you've identified like terms in an expression, the next step is to combine them. This is a straightforward process. You essentially perform the arithmetic operations on the coefficients of the same variable. In the given expression, we found the like terms as \(13x\) and \(23x\), and as well as \(-25y\) and \(-35y\). By combining, you add the coefficients:

• For the \(x\) terms: \(13x + 23x = 36x\)
• For the \(y\) terms: \(-25y - 35y = -60y\)

Notice how we add or subtract only the numerical parts while the variable stays unchanged. This step simplifies the expression significantly and reduces it to fewer terms, making it easier to work with in further operations.

Simplified Expression

After combining the like terms, you've reached an important milestone: the simplified expression. Simplification involves condensing complex expressions into a more manageable form without changing their original value. For our expression, after grouping and combining like terms, we end up with \(36x - 60y\). This is the simplest form of the expression, capturing the same value but in a neat, concise way. Simplified expressions are easier to understand and work with, especially in more complex algebraic problems. This final result can be used seamlessly in equations or functions, allowing you to solve, analyze, and graph them more effectively.