Problem 14 - 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. $$7 x+5 x+2+9$$

Step-by-Step Solution

Answer

The expression simplifies to \(12x + 11\).

Step 1: Identify Like Terms

Look at the expression: \(7x + 5x + 2 + 9\). Identify the terms that are like each other. Like terms are terms that have the same variables raised to the same powers. In this expression, \(7x\) and \(5x\) are like terms, and \(2\) and \(9\) are constant terms.

Step 2: Rearrange Terms Using Commutative Property

Reorganize the expression to group like terms together. The commutative property states that you can add terms in any order. Thus, rearrange to \((7x + 5x) + (2 + 9)\).

Step 3: Combine the Like Terms

Add the coefficients of the like terms. For \(7x + 5x\), add the coefficients: \(7 + 5 = 12\), so it becomes \(12x\). For the constants, add \(2 + 9 = 11\). Combine them to get \(12x + 11\).

Key Concepts

Commutative Property

The commutative property is a fundamental rule in mathematics that helps us rearrange terms in an expression without affecting its overall value. Simply put, this property allows us to change the order of the terms when we are adding or multiplying. For example, if we have numbers or terms such as \(a\) and \(b\), the commutative property tells us that \(a + b = b + a\) or \(a \cdot b = b \cdot a\).
In the context of simplifying expressions, using this property means you can group similar terms together to make the calculation easier. As seen in the exercise, we rearranged the terms \(7x + 5x + 2 + 9\) to \((7x + 5x) + (2 + 9)\) so that like terms are next to each other. This step is crucial in simplifying expressions as it sets the stage for combining like terms efficiently.

Like Terms

Like terms are an important concept in algebra which refers to terms that have the same variable raised to the same power. This means they can be combined through addition or subtraction because they represent the same quantity. In the problem \(7x + 5x + 2 + 9\), the terms \(7x\) and \(5x\) are like terms because they both contain the variable \(x\) to the first power.
Distinguishing like terms from others is essential for simplifying an expression. The coefficients (numerical parts) of like terms can be directly added or subtracted. For example, in our exercise, the coefficients 7 and 5 are added together to yield \(12x\). On the other hand, constant terms, like 2 and 9 in our case, can also be grouped and simplified because they do not have variables. Identifying and combining like terms is a straightforward yet powerful step in making complex algebraic expressions more manageable.

Constant Terms

Constant terms are the standalone numbers in an expression that do not contain any variables. They are crucial to recognize because, unlike variable terms, they can be combined only with other constant terms. In the expression \(7x + 5x + 2 + 9\), the constant terms are 2 and 9.
If you treat constants like their own group of like terms, you can simplify them by adding them together, just like how we deal with variable terms. In our exercise, adding the constant terms 2 and 9 results in 11. This makes it easier to manage parts of the expression without affecting other variable components. Constant terms might seem simple, but keeping them in check is key to ensuring we've truly simplified the entire expression.