Verified Answer :

  • Group like terms: \((5y + 3y)\) and constants \((-4 + 10)\).
  • Add the coefficients of the \(y\) terms: \(5y + 3y = 8y\).
  • Combine the constant terms: \(-4 + 10 = 6\).
  • Final simplified expression: \(\boxed{8y + 6}\).

Explanation

  1. Understanding the Problem

The goal is to simplify the expression \((5y - 4) + (3y + 10)\). This expression consists of two linear polynomials in the variable \(y\). To simplify it, we need to group the like terms together. Like terms are terms that have the same variable raised to the same power, as well as constant terms that have no variable.

  1. Grouping Like Terms

Using the commutative and associative properties of addition, we can rearrange the terms of the expression. The terms containing the variable \(y\) are \(5y\) and \(3y\). The constant terms are \(-4\) and \(10\). We can group them as follows:
\[(5y - 4) + (3y + 10) = (5y + 3y) + (-4 + 10)\]

  1. Performing the Addition

Now, we perform the arithmetic for each group separately. For the terms containing \(y\), we add their coefficients:
\[5y + 3y = (5 + 3)y = 8y\]
For the constant terms, we add them together:
\[-4 + 10 = 6\]
Combining these two results, we get the simplified expression:
\[8y + 6\]

Examples

Simplifying algebraic expressions is fundamental in many real-life applications. For example, if you are calculating the total cost of a shopping trip where you buy 5 items that cost \(\$y\) each minus a \(\$4\) discount coupon, and then buy 3 more items at \(\$y\) each plus a \(\$10\) shipping fee, you can use algebra to find the total: \((5y - 4) + (3y + 10) = 8y + 6\). This simplifies the calculation so you only have to perform one operation based on the price of the items.

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