Problem 57 - Prealgebra

Question

Simplify each expression. $$3-2(y+4)$$

Step-by-Step Solution

Answer

The simplified expression is -5 - 2y.

Step 1: Distribute the Negative Sign

Take the expression inside the parentheses and multiply each term by the factor outside, which is -2. This gives -2(y) + -2(4).

Step 2: Simplify the Expression

Now, simplify the expression. This becomes 3 - 2y - 8. Combine the like terms, which are the constants: 3 - 8 = -5.

Step 3: Write the Final Expression

Combine the constants to get the simplified expression as -5 - 2y.

Key Concepts

Simplifying Expressions

When you simplify an expression, your goal is to make it as easy as possible. Simplification doesn't change the value of the expression; it just makes it more direct.

Imagine you have a messy room and cleaning it makes everything easier to find. That's what simplifying does to an expression.

• Look for parts of the expression that can be combined or rewritten in simpler form.
• Each step should reduce the complexity.
• A simplified expression helps to understand and solve algebraic equations more intuitively.

Once simplified, expressions like our example i.e., \(3 - 2(y + 4)\), become much easier to manage in further calculations and problem-solving.

Distributive Property

The distributive property allows you to multiply a number by a sum or difference inside parentheses. It states that multiplying a number by a group of numbers added together is the same as doing it separately.

In our example, the expression \(-2(y + 4)\) requires you to distribute -2 to both \(y\) and \(4\).

• Apply the negative outside the parentheses to each term inside.
• This leads to \(-2 \cdot y\) and \(-2 \cdot 4\).

This step transforms the initial problem into a more straightforward arithmetic form, making the calculation easier in later steps.

Combining Like Terms

Combining like terms is another essential skill in simplifying expressions. Terms are considered 'like' if they have the same variables raised to the same power.

In our example, once the distributive property has been applied to \(-2y - 8\), you identify terms that can be added together.

• Constants are \(3\) and \(-8\): Combine them.
• This gives \(-5\) (the sum of \(3\) and \(-8\)).

Combining like terms reduces the amount of work you have to do and simplifies the equation to its most compact form.

Prealgebra

Prealgebra is the stepping stone to more advanced mathematics, introducing students to concepts like variable manipulation and simple equations.

Focusing on fundamentals, like understanding how to handle negative numbers and the distributive property, builds a strong base.

• Learn to deal with variables alongside numbers.
• Apply properties like distribution to simplify expressions early.

Mastering prealgebra concepts ensures you are prepared to tackle algebraic structures encountered in higher-level math, including the expression \(-5 - 2y\), simplified from \(3 - 2(y + 4)\).