Problem 66 - College Algebra

Question

simplify each algebraic expression. $$ 4(5 y-3)-(6 y+3) $$

Step-by-Step Solution

Answer

The simplified form of the given algebraic expression is \(14y - 15\).

Step 1: Expand the expression

Begin by expanding the brackets by multiplying the quantity inside the bracket with the quantity outside the bracket. Doing this, the equation turns into \(20y - 12 - 6y - 3\).

Step 2: Combine like terms

The next step is to combine like terms in the expression. This would result in the combination of \(20y\) and \(-6y\) and separately, \(-12\) and \(-3\) giving \(14y - 15\).

Key Concepts

Simplifying Expressions

When we talk about simplifying expressions in algebra, we're essentially looking at turning a complex-looking expression into a simpler form. The aim here is to make it as concise and tidy as possible while retaining the same mathematical meaning. This often involves a combination of expanding expressions and combining like terms. Think of it as tidying up a messy room. You don’t change the room, but you make it look nicer. In the original problem, this involves taking a complex expression and reducing it down to a cleaner form. By the end of the process, your expression is left with fewer operations and terms, but it packs the same mathematical punch.

Expanding Expressions

In algebra, expanding expressions means removing parentheses by multiplying the terms inside the parentheses by whatever is outside them. This is also known as the distributive property. For example, consider the expression \(4(5y - 3)\). Here, 4 needs to be distributed to both \(5y\) and \(-3\).

• This means you multiply 4 by \(5y\), giving \(20y\).
• Then, multiply 4 by \(-3\), to get \(-12\).

Combining these, the expression \(4(5y - 3)\) simplifies to \(20y - 12\). In the example given, the same logic of expansion is applied through the entire expression, turning it into \(20y - 12 - 6y - 3\).

Expanding expressions simplifies later stages of solving since it lays out all terms in a straightforward manner without any parentheses to worry about.

Combining Like Terms

Combining like terms is usually the last step in simplifying algebraic expressions. Like terms are those that contain the same variable raised to the same power. They're like twins in a math expression—they look alike and can be combined into one.

Consider the expression from our example after expansion: \(20y - 12 - 6y - 3\). Here we identify the like terms:

• The \(20y\) and the \(-6y\) are like terms since they both involve the variable \(y\).
• The \(-12\) and \(-3\) are like terms as they are both constants.

Now, we sum the like terms:

• Add \(20y\) and \(-6y\) to get \(14y\).
• Then, combine \(-12\) and \(-3\) to get \(-15\).

Therefore, the expression simplifies to \(14y - 15\).

Combining like terms helps to further simplify the expression, leaving it in its simplest form and making it easier to work with for any additional algebraic operations.