Problem 12 - Prealgebra

Question

Simplify expression. \(5-3(y+7)\)

Step-by-Step Solution

Answer

The simplified expression is \(-3y - 16\).

Step 1: Distribute the Negative Sign

Start by distributing the negative sign in the expression \(5 - 3(y + 7)\). This involves multiplying \(-3\) with both \(y\) and \(7\). This will change the expression to \( 5 - 3y - 21 \).

Step 2: Combine Like Terms

Now, combine the constants in the simplified expression from the previous step. You have \( 5 - 21\), which simplifies to \(-16\). Therefore, the expression becomes \(-3y - 16\).

Key Concepts

Simplifying Expressions

Simplifying expressions means making them easier to work with by performing operations and reducing their complexity. For the problem \(5 - 3(y + 7)\), it involves using the distributive property and combining like terms.
When you simplify an expression:

• Break down the terms to their simplest forms
• Use mathematical operations such as addition, subtraction, and multiplication
• Remove any unnecessary components to make the expression easier to understand

In the given expression, first distribute the multiplication over addition within the parentheses. Multiply \(-3\) with both \(y\) and \(7\), which results in the expression \(5 - 3y - 21\). After this, move to combining like terms to further simplify the expression.

Combining Like Terms

Combining like terms is a crucial step in simplifying algebraic expressions. Like terms are terms that have the same variables raised to the same power. For example, terms with \(y\) can be combined with other terms that contain \(y\).
In the expression \(5 - 3y - 21\):

• 5 and -21 are constants, which means they are like terms and can be combined
• The \(-3y\) term involves the variable \(y\), and must stand alone unless there are other \(y\)-terms to combine with it

To combine the constants 5 and -21, you add them together: \(5 - 21 = -16\). As a result, your expression becomes \(-3y - 16\). This approach simplifies the algebraic expression to make it clearer and more compact.

Negative Numbers

Negative numbers represent a value less than zero and can change the results of operations significantly. When simplifying expressions, it’s important to handle negative numbers with care. In our example, the negative number is used in the distribution step.
The expression \(5 - 3(y + 7)\) requires distributing the \(-3\) across both terms inside the parentheses:

• Multiply \(-3\) by \(y\) to get \(-3y\)
• Multiply \(-3\) by \(7\) to get \(-21\)

These steps result in the expression \(5 - 3y - 21\). Understanding negative numbers is crucial because:

• They can alter the signs of terms in expressions
• A negative times a positive gives a negative product
• Combining negative numbers requires careful addition or subtraction

By accurately managing negative numbers, you ensure the simplified expression maintains its correct value.