Problem 65 - College Algebra

Question

simplify each algebraic expression. $$ 5(3 y-2)-(7 y+2) $$

Step-by-Step Solution

Answer

The simplified form of the given algebraic expression is \(8y - 12\).

Step 1: Apply Distributive Property

Distribute the \(5\) across the \(3y - 2\) and the negation signal across the \(7y + 2\). This should be done as follows: \(5 * 3y - 5*2 - 1*7y - 1*2\) giving \(15y -10 -7y - 2\)

Step 2: Combine Like Terms

Now, group the 'y' terms together and the constant terms together to simplify the expression. This can be done as follows: \( (15y - 7y) + ( -10 - 2)\) to get \(8y - 12\)

Step 3: Final Simplification

The expression has already been reduced to its simplest form, \(8y - 12\), there are no more like terms to combine.

Key Concepts

Distributive Property

Understanding the distributive property is key to simplifying many algebraic expressions. It allows you to multiply a single term by each term inside a parenthesis. Imagine distributing a bag of cookies evenly among your friends; you're giving an equal amount to each person. Similarly, in algebra, you're giving the single term outside the parenthesis to every term inside. For example, take the expression \(5(3y-2)\). Here, you distribute the \(5\) across both \(3y\) and \(-2\), resulting in \(5 \times 3y\) and \(5 \times -2\), which gives you \(15y - 10\).

The same principle is applied when you have a subtraction sign before the parenthesis, such as \(-(7y + 2)\). This is equivalent to \(-1(7y + 2)\), so you distribute the \(-1\) across \(7y\) and \(+2\), getting \(-7y - 2\). This step is crucial as it breaks down the expression into individual terms that can be simplified further.

Combine Like Terms

After distributing, the next step is to combine like terms. Like terms are terms in an algebraic expression that have the same variable raised to the same power. Think of it like grouping similar items together when you clean your room; socks with socks, shirts with shirts. In the expression from our example, \(15y\) and \(-7y\) are like terms because they both contain the variable \(y\). When you combine them, you add their coefficients, the numbers in front of the variables, to get \(15y - 7y\), which simplifies to \(8y\).

Similar to variables, constants (numbers without variables) are also combined. In our case, the constants are \(-10\) and \(-2\). You add these together to get \(-10 - 2\), resulting in \(-12\). Combining like terms is a fundamental skill in algebra that simplifies expressions and makes them easier to handle in further calculations.

Algebraic Simplification

Algebraic simplification is the process of reducing an algebraic expression to its simplest form. This is done after applying the distributive property and combining like terms. It's akin to solving a puzzle; you simplify complex pieces step by step until you see the complete picture. In our example, we reached \(8y - 12\) after combining like terms. At this stage, no further simplification is possible as there are no more like terms to combine, making \(8y - 12\) the simplest form of our original expression.

This doesn't always mean that the expression becomes a single term, but that you've done all you can to make the expression as uncomplicated as possible. Simplification makes algebraic expressions cleaner and more straightforward, which is particularly helpful when solving equations and inequalities where clarity is vital.