Problem 58 - Prealgebra

Question

You use deductive reasoning when you base a conclusion on mathematical rules or properties. Indicate the property that justifies each step that was used to simplify \(3(x+4)+5(x+1)\). \(=8 x+17\)

Step-by-Step Solution

Answer

The expression simplifies to \(8x + 17\) using the distributive property and combining like terms.

Step 1: Apply the Distributive Property

In the expression \(3(x+4)+5(x+1)\), apply the distributive property which states that \(a(b+c) = ab + ac\). Distributing 3 and 5 to their respective terms inside the parentheses, we get: \(3 \times x + 3 \times 4 + 5 \times x + 5 \times 1\). This simplifies to \(3x + 12 + 5x + 5\).

Step 2: Combine Like Terms

Now that we have \(3x + 12 + 5x + 5\), we need to combine the like terms. The like terms are the terms with \(x\) and the constant terms. Combine \(3x\) and \(5x\) to get \(8x\), and combine \(12\) and \(5\) to get \(17\). Thus, the simplified expression is \(8x + 17\).

Key Concepts

Commutative Property

Understanding the Commutative Property is crucial in simplifying expressions. This property states that when you add or multiply, the order of the numbers does not change the result. For addition, the commutative property can be expressed as \(a + b = b + a\). Similarly, for multiplication, it is \(a \times b = b \times a\). These principles are powerful when you're rearranging terms in an expression to make it easier to work with.

Imagine you have the expression \(3x + 5x\). By the commutative property, you can swap \(3x\) and \(5x\) to get \(5x + 3x\). The total does not change, and this flexibility can simplify calculations. Always remember, whether adding numbers or multiplying them, the commutative property allows the rearrangement of terms to your advantage which can make further steps, such as combining like terms, more intuitive.

Combine Like Terms

Combining like terms speeds up the simplification of expressions. Like terms are terms that have the same variable raised to the same power. For instance, in the expression \(3x + 5x + 12 + 5\), \(3x\) and \(5x\) are like terms because they both contain the variable \(x\), and \(12\) and \(5\) are like terms as they are constant terms.

When you combine like terms, you're essentially adding or subtracting the coefficients of the terms while keeping the variable part the same. In our expression, \(3x + 5x\) combines to \(8x\), and \(12 + 5\) combines to \(17\). This reduces the expression into a simpler form: \(8x + 17\).

• First, identify terms with the same variables.
• Next, group them together.
• Finally, perform the addition or subtraction of the coefficients.

By neatly tucking together like terms, expression simplification becomes a breeze.

Expression Simplification

Simplifying an expression involves reducing it to its most straightforward form. This process makes calculations easier and solutions more apparent. It often involves using various mathematical properties and techniques such as the distributive property, commutative property, and combining like terms.

Starting with complex expressions, see if you can apply properties like distribution to break down terms. For example, in \(3(x + 4) + 5(x + 1)\), use the distributive property to get \(3x + 12 + 5x + 5\). Then apply the commutative property to rearrange and group like terms together: \(3x + 5x + 12 + 5\).

Afterward, use the knowledge of combining like terms to sum up the coefficients: \(8x + 17\). This journey from a lengthy expression to a neatly wrapped \(8x + 17\) illustrates the power of simplification.

• Follow the steps in order: distribute, rearrange, then combine.
• Ensure each step is done with purpose and clarity.
• This approach helps maintain organization and ensures no term is overlooked.

Simplifying expressions not only enhances mathematical skills but also builds a strong foundation for tackling more complex problems in mathematics.