Problem 50 - Elementary Algebra Within Reach

Question

In Exercises \(47-66\), simplify the expression by removing symbols of grouping and combining like terms. $$ (2 x-1) 2+x+9 $$

Step-by-Step Solution

Answer

The simplified expression is \(5x + 7\).

Step 1: Apply Distributive Property

The first part in the equation is \((2x - 1)\) multiplied by \(2\). This is a perfect instance to apply the distributive property of multiplication over addition. Applying the distributive property, \(2 * (2x - 1)\) becomes \(4x - 2\). The expression thus becomes: \[ 4x - 2 + x + 9 \]

Step 2: Combine Like Terms

The next step is to combine like terms. 'Like terms' refer to those terms which have the same variable(s) raised to the same power(s). Here, the like terms are \(4x\) and \(x\) as well as -2 and 9. When added together, they simplify to: \[ 5x + 7 \]

Key Concepts

Distributive Property

When we talk about simplifying algebraic expressions, one of the most helpful tools at our disposal is the distributive property. It states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. In practice, this looks like: \(a(b + c) = ab + ac\).

Let's apply this to our exercise. Given \(2x - 1\) multiplied by 2, we distribute the 2 across the terms inside the parentheses, which gives us \(4x - 2\). Here's a simple breakdown:

• Multiply the first term inside the parenthesis \(2x\) by 2 to get \(4x\).
• Then, multiply the second term \(–1\) by 2 to get \(–2\).

This process streamlines the expression, making it ready for further simplification.

Combining Like Terms

Once we've distributed and removed symbols of grouping, our next step is to scrutinize the expression for like terms. Like terms have identical variables raised to the same power. Their coefficients (the numbers in front of the variables) can be different, but as long as the variable parts match, they can be combined.

In our example, we encountered the terms \(4x\) and \(x\). Here’s how you would go about combining them, and why:

• Identify the like terms—here, \(4x\) and \(x\)—which both have the variable \(x\) raised to the first power.
• Add their coefficients. The term \(x\) is actually \(1x\), so \(4x + 1x = 5x\).

Similarly, combine the numbers without variables, \(–2\) and \(9\), which gives us \(7\). Ultimately, we get the simplified expression \(5x + 7\).

Symbols of Grouping

Algebraic expressions often include symbols of grouping like parentheses \( ( ) \), brackets \[ [ ] \], or braces \( \{ \} \). These symbols indicate the parts of the expressions that should be considered as a single unit. When simplifying expressions, it’s crucial to deal with these groups appropriately.

In the provided exercise, we had parentheses around \(2x - 1\). Our job is to simplify what’s inside them first, or, if there is a value outside like the number 2, to apply the distributive property to remove the parentheses.

• Inspect the expression for any symbols of grouping.
• Perform operations inside the grouping symbols or use the distributive property if there is a multiplication sign outside the grouping symbol.

Once the symbols of grouping are removed, like in our step by step solution, the expression can then be re-evaluated to combine like terms and simplify further.