Problem 28 - 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. $$7 x+5-2 x+6$$

Step-by-Step Solution

Answer

The expression simplifies to \( 5x + 11 \).

Step 1: Identify Like Terms

First, observe the expression to identify terms that can be combined. The expression is given as \( 7x + 5 - 2x + 6 \). The like terms are the ones involving \( x \), namely \( 7x \) and \( -2x \). Similarly, the constant terms are \( 5 \) and \( 6 \).

Step 2: Rearrange using the Commutative Property

Rearrange the terms to group like terms together using the commutative property, which allows us to change the order of terms. Rearranging the expression gives us \( 7x - 2x + 5 + 6 \).

Step 3: Combine Like Terms Involving x

Combine the like terms involving \( x \). Here, we add \( 7x \) and \( -2x \) to get: \( (7 - 2)x = 5x \).

Step 4: Combine Constant Terms

Now, combine the constant terms \( 5 \) and \( 6 \). Add these numbers together to get \( 5 + 6 = 11 \).

Step 5: Write the Simplified Expression

Combine the results from the previous steps to form the final simplified expression: \( 5x + 11 \).

Key Concepts

Like Terms

When you're simplifying an expression in algebra, it's important to first understand what like terms are. Like terms are terms in an expression that contain the same variable raised to the same power. For example, in the expression \(7x + 5 - 2x + 6\), the terms \(7x\) and \(-2x\) are like terms because they both contain the variable \(x\) raised to the first power.

Likewise, constant terms are like terms as well, because they do not contain any variables. This means you can combine them just like any other like terms. In our example, \(5\) and \(6\) are constants, so they can be combined.

When simplifying expressions, always look for like terms as they can significantly make the expression shorter and easier to handle.

Commutative Property

The commutative property is a fundamental rule in mathematics that allows you to rearrange terms in an expression without changing the overall value. This property applies to both addition and multiplication and is extremely useful when working with expressions involving like terms.

For example, in the expression \(7x + 5 - 2x + 6\), you can rearrange it to \(7x - 2x + 5 + 6\) using the commutative property of addition. This makes it easier to group and subsequently combine the like terms together. Remember, this property states that \(a + b = b + a\) and \(ab = ba\), so you're free to change the order of terms as needed to simplify your expression.

Combining Terms

Once like terms are identified and rearranged using the commutative property, the next step is to actually combine these terms. Combining terms means performing the addition or subtraction of the coefficients (the numbers in front of variables) of like terms, as well as adding constant terms.

In the expression \(7x - 2x + 5 + 6\), you would combine the coefficients of \(x\) terms like this: \((7 - 2)x = 5x\). Then, you would combine the constant terms by adding them together: \(5 + 6 = 11\).

After combining the terms, the expression cleverly simplifies to \(5x + 11\), which is significantly easier to work with than the original. It's like neatly packing an overstuffed carry-on bag; everything fits much better when you've organized it properly.