Problem 44 - Elementary Algebra

Question

Simplify. (Assume all denominators are nonzero.) $$ 3 x-1-2 x+32 x+3+1 x-3 $$

Step-by-Step Solution

Answer

The simplified expression is \(34x - 1\).

Step 1: Identify Like Terms

In the expression \(3x - 1 - 2x + 32x + 3 + 1x - 3\), identify terms that have \(x\) and constant terms separately. Here, the terms with \(x\) are \(3x\), \(-2x\), \(32x\), and \(1x\). The constants are \(-1\), \(3\), and \(-3\).

Step 2: Combine Like Terms with \(x\)

Add the coefficients of the \(x\) terms: \(3x - 2x + 32x + 1x = (3 - 2 + 32 + 1)x = 34x\).

Step 3: Combine Constant Terms

Perform the addition of the constants: \(-1 + 3 - 3 = -1\).

Step 4: Write Simplified Expression

Combine the results from steps 2 and 3 to write the simplified expression: \(34x - 1\).

Key Concepts

Like Terms

When working with algebraic expressions, understanding like terms is crucial for simplifying the expression. Like terms are terms that have the same variable raised to the same power. Only the numerical coefficients of these terms can differ. For instance, in the expression \(3x + 2x - x\), all terms involve the variable \(x\) raised to the power of one. This makes them like terms.
Breaking down the expression, we see that:

• \(3x\) and \(2x\) each have \(x\) as the variable.
• \(-x\) or \(-1x\) also shares the same variable, making it a like term.

To simplify, you would add or subtract the coefficients of these like terms while keeping the variable intact, resulting in a simpler form. This concept can be applied to more complex expressions provided the terms share both variable and exponent similarities.

Combining Constants

Constants are numbers on their own, without any attached variables. In simplifying expressions, combining constants is a straightforward yet essential part. Consider the expression: \(-1 + 3 - 3\). These are constants without any variables, making them entirely numerical values.
When simplifying, you add or subtract these constants as you would regular numbers:

• First, add \(3\) and \(-1\) to get \(2\).
• Next, subtract \(3\) from \(2\) to arrive at \(-1\).

The key to combining constants is keeping track of signs, as these determine whether you add or subtract the values in question. This simplified result, \(-1\), represents the combined value of the original constants in the expression.

Coefficients in Algebra

Coefficients are the numerical parts of algebraic terms, and they play a pivotal role in simplifying expressions. In an expression like \(3x - 2x + 32x + 1x\), the coefficients are \(3\), \(-2\), \(32\), and \(1\), accompanying the variable \(x\).
To simplify using the coefficients, follow these steps:

• Identify all terms with the same variable.
• Add or subtract the coefficients to simplify.

For this example:

• Add \(3\) and \(-2\) to get \(1\).
• Then add \(32\) to \(1\), resulting in \(33\).
• Finally, add \(1\) to \(33\), giving \(34\).

The simplified expression is \(34x\). Understanding how to work with coefficients allows for manipulation and simplification of more complex expressions with ease and clarity.