Simplify 1.8h - 0.7h + P - 6p By Combining Like Terms

JU07/09/2025 08, 2025 by THE IDEN 54 views

Simplify the Expression by Combining Like Terms

In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to reduce complex expressions into a more manageable form, making them easier to understand and work with. One of the key techniques in simplifying expressions is combining like terms. Like terms are terms that have the same variable raised to the same power. By combining these terms, we can streamline expressions and reveal their underlying structure. In this article, we will delve into the process of simplifying expressions by combining like terms, using the expression $1.8h - 0.7h + p - 6p$ as our example.

Understanding Like Terms

Before we can effectively combine like terms, it's crucial to have a solid grasp of what they are. As mentioned earlier, like terms are terms that share the same variable raised to the same power. The coefficient, which is the numerical factor multiplying the variable, can be different. For instance, in the expression $3x + 5x - 2y$ , the terms $3x$ and $5x$ are like terms because they both have the variable $x$ raised to the power of 1. However, the term $-2y$ is not a like term because it has a different variable, $y$ . To solidify your understanding, let’s consider a few more examples:

$7a$ and $-2a$ are like terms.
$4b^2$ and $9b^2$ are like terms.
$6xy$ and $-3xy$ are like terms.
$5z$ and $5z^2$ are not like terms (different powers).
$2p$ and $2q$ are not like terms (different variables).

Identifying like terms is the first step in simplifying expressions. Once you can confidently recognize them, the process of combining them becomes straightforward. Now that we have a clear understanding of like terms, let’s move on to the next step: identifying the like terms in our expression, $1.8h - 0.7h + p - 6p$ .

Identifying Like Terms in the Expression

Now, let's apply our understanding of like terms to the expression we aim to simplify: $1.8h - 0.7h + p - 6p$ . Our goal is to identify the terms that share the same variable raised to the same power. Looking at the expression, we can see two distinct groups of like terms:

Terms with the variable h: We have $1.8h$ and $-0.7h$ . Both of these terms contain the variable $h$ raised to the power of 1, making them like terms.
Terms with the variable p: We have $p$ and $-6p$ . These terms both contain the variable $p$ raised to the power of 1, so they are also like terms.

Notice that the terms with $h$ cannot be combined with the terms with $p$ . They are distinct groups of like terms and must be treated separately. Once we have identified the like terms, we can proceed to the next crucial step: combining these terms by adding or subtracting their coefficients.

Combining Like Terms: The Mechanics

Combining like terms is essentially the process of adding or subtracting the coefficients of those terms while keeping the variable and its exponent the same. The coefficient is the numerical part of the term. For example, in the term $5x$ , the coefficient is 5. In the term $-3y^2$ , the coefficient is -3.

To combine like terms, we follow these simple steps:

Identify the like terms (as we did in the previous section).
Add or subtract the coefficients of the like terms. Remember to pay attention to the signs (positive or negative) of the coefficients.
Keep the variable and its exponent the same. The variable and its exponent do not change during the process of combining like terms.

Let’s illustrate this with a simple example. Suppose we have the expression $4x + 7x$ . Both terms are like terms because they have the variable $x$ raised to the power of 1. To combine them, we add their coefficients:

$4 + 7 = 11$

So, the simplified expression is $11x$ . We added the coefficients (4 and 7) and kept the variable $x$ the same. Now, let’s apply this technique to the expression we want to simplify, $1.8h - 0.7h + p - 6p$ .

Simplifying the Expression: A Step-by-Step Approach

Now that we have a firm understanding of like terms and how to combine them, let's apply this knowledge to simplify the expression $1.8h - 0.7h + p - 6p$ . We'll proceed step by step to ensure clarity.

Step 1: Identify the Like Terms

As we established earlier, the like terms in this expression are:

$1.8h$ and $-0.7h$ (terms with the variable $h$ )
$p$ and $-6p$ (terms with the variable $p$ )

Step 2: Combine the 'h' Terms

To combine the $h$ terms, we add their coefficients:

$1.8 - 0.7 = 1.1$

So, $1.8h - 0.7h$ simplifies to $1.1h$ .

Step 3: Combine the 'p' Terms

Next, we combine the $p$ terms. Remember that when a variable stands alone, it's understood to have a coefficient of 1. So, $p$ is the same as $1p$ . Now, we add the coefficients:

$1 - 6 = -5$

Therefore, $p - 6p$ simplifies to $-5p$ .

Step 4: Write the Simplified Expression

Now that we've combined the like terms, we can write the simplified expression by combining the results from Step 2 and Step 3:

$1.1h - 5p$

This is the simplified form of the original expression, $1.8h - 0.7h + p - 6p$ . We have successfully combined the like terms and reduced the expression to its simplest form.

The Significance of Simplifying Expressions

Simplifying expressions is not just a mathematical exercise; it's a crucial skill with far-reaching implications. Here are some reasons why simplifying expressions is so important:

Clarity: Simplified expressions are easier to understand and interpret. They reveal the underlying structure of the expression, making it clear how the different parts relate to each other.
Efficiency: Working with simplified expressions is much more efficient. They require fewer calculations and reduce the chances of making errors.
Problem-Solving: Many mathematical problems require simplifying expressions as an intermediate step. Whether you're solving equations, graphing functions, or working with complex formulas, simplifying expressions is often necessary.
Advanced Mathematics: The ability to simplify expressions is essential for success in higher-level mathematics courses, such as algebra, calculus, and differential equations. These courses build upon the foundation of algebraic manipulation, and a strong understanding of simplifying expressions is critical.
Real-World Applications: Simplifying expressions has applications in various real-world fields, such as engineering, physics, computer science, and economics. In these fields, complex mathematical models are often used to describe and analyze phenomena, and simplifying these models is crucial for making predictions and solving problems.

In conclusion, mastering the art of simplifying expressions by combining like terms is a valuable investment in your mathematical journey. It not only enhances your problem-solving skills but also lays the groundwork for future success in mathematics and related fields.

Conclusion

In this article, we've explored the process of simplifying expressions by combining like terms. We've learned that like terms are terms with the same variable raised to the same power, and combining them involves adding or subtracting their coefficients. By following a step-by-step approach, we successfully simplified the expression $1.8h - 0.7h + p - 6p$ to $1.1h - 5p$ . We also emphasized the significance of simplifying expressions in mathematics and its applications in various fields. Mastering this skill will undoubtedly prove beneficial as you continue your mathematical journey. Remember, practice makes perfect, so keep simplifying expressions and honing your skills!