Question

Simplify the expression. $$5(y+3)+7 y$$

Short answer

The simplified form of the expression is \(12y + 15\)

Step-by-step solution

Distribute

Distribute the number 5 across the terms inside the parentheses: \(5(y) + 5(3) + 7y\). It simplifies to \(5y + 15 + 7y\).

Combine like terms

Combine the similar terms ('y' terms) in the expression resulting from the first step, i.e., \(5y + 7y + 15\). This simplifies to \(12y + 15\).

Key concepts

Distributive Property

Understanding the distributive property is essential in algebraic expressions, especially when dealing with parentheses. This property allows us to remove parenthetical parts by distributing a multiplier to each term inside the parentheses.

For example, in the expression \(5(y+3)\), we apply the distributive property: the number outside the parentheses (in this case, 5) is multiplied with each term inside the parentheses. So, it becomes \(5\cdot y + 5\cdot 3\).

Here are some key points to remember:

• Each term inside the bracket is multiplied by the number outside.
• The operation ensures that the multiplier is evenly distributed across all terms.
• This essential algebra rule helps simplify complex expressions and solve equations.

Practicing the distributive property will build a strong foundation for more advanced algebra concepts.

Combine Like Terms

When we simplify algebraic expressions, combining like terms is a crucial step. Like terms are terms that contain the same variable raised to the same power. They are the 'apples' or 'oranges' in our algebraic fruit salad - similar items that can be added together.

The expression from our example, \(5y + 15 + 7y\), has two like terms: \(5y\) and \(7y\). We combine them by adding their coefficients (the numbers in front of the 'y'), which gives us \(12y\). The constant \(15\) remains as is because there isn’t another constant to combine it with.

Key tips when combining like terms:

• Identify terms with the same variables and exponents.
• Add or subtract coefficients of like terms.
• Keep the variable and exponent of the like terms unchanged.

Mastering this process helps in simplifying and solving more complex equations.

Algebraic Simplification

Algebraic simplification is the process of making an algebraic expression easier to understand and work with. It involves using various algebra rules, such as the distributive property and combining like terms, as well as performing operations like addition and subtraction correctly.

In our exercise, after distributing and combining like terms, we arrive at the simplified form \(12y + 15\). This expression is easier to evaluate or further manipulate in problem-solving. Simplifying expressions can also reveal properties of the algebraic equation that may not be initially obvious.

As you simplify algebraic expressions, keep these strategies in mind:

• Perform operations in the correct order.
• Reduce fractions where possible.
• Factor when necessary to simplify expressions or solve equations.

Approaching algebraic expressions methodically will lead to success in algebra and beyond.