Problem 17 - Prealgebra

Question

Simplify the following expressions by combining similar terms. In some cases the order of the terms must be rearranged first by using the commutative property. $$-2 a+a+7+5$$

Step-by-Step Solution

Answer

The simplified expression is \(-a + 12\).

Step 1: Identify Like Terms

First, identify terms that can be combined because they have the same variable or no variable at all. In the expression \(-2a + a + 7 + 5\), the like terms are the variable terms \(-2a\) and \(a\), and the constant terms \(7\) and \(5\).

Step 2: Rearrange the Expression

Use the commutative property to rearrange the terms so that like terms are together: \(-2a + a + 7 + 5\) becomes \((-2a + a) + (7 + 5)\).

Step 3: Combine Like Terms (Variable Terms)

Add the coefficients of the variable terms: \((-2 + 1)a = -1a\). So the variable terms simplify to \(-a\).

Step 4: Combine Like Terms (Constant Terms)

Add the constant terms: \(7 + 5 = 12\).

Step 5: Write the Simplified Expression

Combine the results from Steps 3 and 4 to get the final simplified expression: \(-a + 12\).

Key Concepts

Combining Like Terms

In prealgebra, combining like terms is a fundamental skill that helps simplify mathematical expressions. Imagine you have different colored blocks; combining like terms is like grouping the same colored blocks together.

In algebraic expressions, terms can be constants (like numbers) or variables (like "a"). When you see terms like \(-2a\) and \(a\), they can be combined because they both involve the variable "a." The coefficients are the numbers in front, and these are what we combine. So, for \(-2a + a\), we look at the numbers \(-2\) and \(+1\) and then calculate \(-2 + 1 = -1\). Thus, these terms combine into \(-a\).

Similarly, constant terms, which are just numbers without variables, can be added together. In the expression given, \(7\) and \(5\) are constants, and their sum is \(12\). By combining like terms efficiently, you make expressions much easier to work with.

Commutative Property

The commutative property is a helpful rule that allows us to rearrange numbers and terms in expressions and equations. It states that terms can be added or multiplied in any order without changing their combined value.

For addition, this means:

• \(a + b = b + a\)

In simpler terms, \(-2a + a + 7 + 5\) doesn’t change its value, even if you rearrange it to \((-2a + a) + (7 + 5)\).

Using this property helps us group all like terms together conveniently so they can be combined with ease. By rearranging terms, we set up our expression in an understandable way, allowing for the combination of similar terms to become straightforward and effortless.

Simplifying Expressions

Simplifying expressions involves taking a complex-looking equation and reducing it to its simplest form. This makes it easier to understand and to solve further if necessary.

When simplifying, you aim to:

• Identify like terms
• Use properties like the commutative property to rearrange terms
• Combine like terms until you reach a state where no further combination is possible

In our given expression \(-2a + a + 7 + 5\), we first use the commutative property to rearrange and group like terms. Then, we perform arithmetic operations, combining \(-2a\) with \(a\) and \(7\) with \(5\). As a result, the simplified form is \(-a + 12\).

This not only makes the expression clearer but often reveals underlying patterns and relationships between the numbers and variables involved.