Problem 36 - Elementary Algebra

Question

Simplify. (Assume all denominators are nonzero.) $$ 3 x-7+1-2 x x 2 $$

Step-by-Step Solution

Answer

The simplified expression is \(-x - 6\).

Step 1: Simplify the Expression Inside

First, simplify the expression inside the given equation. The equation is \(3x - 7 + 1 - 2x \times 2\). Notice the term \(-2x \times 2\) which can be simplified to \(-4x\).

Step 2: Combine Like Terms

Now, combine like terms. The equation \(3x - 7 + 1 - 4x\) contains two like terms: \(3x\) and \(-4x\). Combine them to get \(-x\).

Step 3: Simplify Constant Terms

Next, combine the constant terms \(-7\) and \(+1\). Adding these, we get \(-6\).

Step 4: Write the Simplified Expression

Combine the results of the previous steps. The simplified expression is \(-x - 6\).

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, one of the key steps involves combining like terms. Like terms are terms in an expression that contain the same variables raised to the same power. For example, in the expression \(3x - 4x\), both terms are "like" because they share the variable \(x\). To combine them, simply add or subtract the coefficients: \(3 - 4 = -1\). So, \(3x - 4x\) simplifies to \(-x\).

• Identify terms with the same variable and power.
• Add or subtract the coefficients.
• Keep the common variable part unchanged.

This step is critical as it helps reduce the complexity of the expression by minimizing the number of separate terms.

Expression Simplification

Simplifying expressions makes them easier to work with by reducing the number of terms and operations. The process generally involves a series of steps starting with removing parentheses or any distributive operations, combining like terms, and then reducing constants. Let's consider the original expression from the exercise: \(3x - 7 + 1 - 2x \times 2\).
First, simplify any operations within the expression. Here, \(-2x \times 2 = -4x\). Next, we combine like terms: \(3x\) and \(-4x\) become \(-x\). Finally, combine the constant terms \(-7 + 1 = -6\).
The expression is now in its simplest form: \(-x - 6\). Simplification achieves a more concise formula, making computations and further algebraic processes more straightforward.

Algebraic Constants

In algebraic expressions, constants are numbers on their own, without any attached variables. These play an essential role in the simplification process. In our working example, the constants are \(-7\) and \(+1\). Unlike terms with variables, constants can be combined just like regular numbers in arithmetic.
Here are key points to handle algebraic constants:

• Look for standalone numbers in the expression.
• Add or subtract them just as you would with simple numbers.
• Ensure numerical accuracy to avoid errors in the final outcome.

In our exercise, adding \(-7\) and \(+1\) results in \(-6\). This step reduces the expression to its most concise form and ensures clarity, especially when performing further algebraic operations.