Problem 70 - Elementary Algebra

Question

Simplify. $$ 3-(2 x+7) $$

Step-by-Step Solution

Answer

The simplified expression is \(-4 - 2x\).

Step 1: Distribute the Negative Sign

The expression given is \(3 - (2x + 7)\). To simplify it, we first distribute the negative sign across the terms inside the parentheses. This means that we will subtract each term inside the parentheses from 3. The expression becomes \(3 - 2x - 7\).

Step 2: Combine Like Terms

Now look at the expression \(3 - 2x - 7\). Notice that there are two constant terms: 3 and -7. Combine these two constants: \(3 - 7 = -4\). So, the expression can be rewritten as \(-4 - 2x\).

Key Concepts

Distribute the Negative Sign

When you encounter an algebraic expression with a negative sign in front of parentheses, the first step is to distribute this negative sign. In simpler terms, distributing means applying the negative sign to each term inside the parentheses.
To do this, change the sign of each term. For instance, in the expression \(3 - (2x + 7)\), you need to:

• Subtract \(2x\), changing \(+2x\) to \(-2x\).
• Subtract 7, changing \(+7\) to \(-7\).

After applying the negative sign to each term, the expression becomes \(3 - 2x - 7\). This ensures that each term inside the bracket is correctly incorporated into the expression without being skipped.
Remember, distributing the negative sign helps in breaking down the problem into simpler parts, setting the stage for the next steps in simplifying expressions.

Combine Like Terms

Once you've distributed any negative signs, the next step is to combine like terms in the expression. But what does that mean?
Like terms are terms that contain the same variable raised to the same power or constants. In the expression \(3 - 2x - 7\), the like terms are the constants \(3\) and \(-7\). These can be combined to simplify the expression further.

• Add or subtract the constants as usual. In this case, calculate \(3 - 7\) which equals \(-4\).

This combination results in the expression being simplified to \(-4 - 2x\). Combining like terms is crucial as it reduces the expression to its simplest form and makes it easier to interpret or use in future calculations.

Simplification of Expressions

Simplifying expressions involves reducing them to their simplest form. This is a handy skill, especially in algebra, as it helps in easily solving and understanding equations.
When simplifying, you follow a systematic approach:

• Distribute factors: Like the negative sign or other factors across terms.
• Combine like terms: Gather and perform operations on similar terms.
• Order of operations: Sometimes you need to apply known mathematical rules like PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) in more complex cases.

For the expression \(3 - (2x + 7)\), we saw these steps in action. We distributed, combined like terms, and ended with \(-4 - 2x\), which is much simpler than our starting point.
Mastering simplification not only boosts your confidence in dealing with algebraic expressions but also prepares you for solving more intricate problems in mathematics.