Problem 12 - Algebra for College Students

Question

Simplify the algebraic expressions in Problems \(1-14\) by combining similar terms. $$5 x-2-7 x+4-x-1$$

Step-by-Step Solution

Answer

The simplified expression is \(-3x + 1\).

Step 1: Identify Like Terms

Like terms in algebraic expressions are terms whose variables and their exponents are the same. In the expression \(5x - 2 - 7x + 4 - x - 1\), group the terms with the variable \(x\): \(5x\), \(-7x\), and \(-x\). The constant terms are \(-2\), \(4\), and \(-1\).

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

Add or subtract the coefficients of the like terms involving \(x\). The expression is: \(5x - 7x - x\). This simplifies to \(-3x\), because \(5 - 7 - 1 = -3\).

Step 3: Combine Constant Terms

Add or subtract the constant terms: \(-2\), \(4\), and \(-1\). This gives \(1\), because \(-2 + 4 - 1 = 1\).

Step 4: Write the Simplified Expression

Combine the results from Step 2 and Step 3 to get the simplified expression: \(-3x + 1\).

Key Concepts

Like Terms

In algebra, simplifying expressions often revolves around the idea of combining like terms. Like terms are terms in the expression that have identical variable parts. These terms can be added or subtracted from each other. For instance, in the expression provided, like terms are the ones having the variable \(x\): \(5x\), \(-7x\), and \(-x\). Each of these terms has \(x\) as the variable and no exponent other than the implicit 1. This shared characteristic allows them to be combined.

Identifying like terms is crucial because it enables you to simplify expressions effectively. To do this, simply look at the variables and their powers. If they match, they're like terms, regardless of their coefficients.

• Like terms must have the same variable component.
• They must also have the same exponent on their variables.
• The coefficients can differ, yet that doesn't affect their status as like terms.

Mastering the art of identifying like terms is a foundational skill in algebra that makes working with expressions much easier.

Combining Coefficients

Once you've identified the like terms in an expression, the next step is to combine their coefficients. Coefficients are the numerical parts of the terms. For example, in the terms \(5x\), \(-7x\), and \(-x\), the coefficients are \(5\), \(-7\), and \(-1\), respectively.

To combine these, you simply add or subtract them as indicated by the signs before them. In the example, combining the coefficients means calculating \(5 - 7 - 1 = -3\). Thus, the result for the variable terms becomes \(-3x\).

• Coefficients are the numbers in front of the variables.
• Signs before coefficients dictate if you add or subtract them.
• Ensure to align calculation based on arithmetic rules.

Accurate combining of coefficients helps simplify expressions and paves the way for easier problem-solving.

Constant Terms

Constant terms are the numbers in an expression that don't have any variable attached to them. These are treated separately from variable terms because they stand independent of variables. In our original expression, these constants are \(-2\), \(4\), and \(-1\).

To simplify them, add or subtract these numbers just like regular arithmetic. Here, combining the constants leads to \(-2 + 4 - 1 = 1\). This final result, \(1\), is part of the simplified expression.

• Constant terms have no variable attached.
• They can be positive or negative.
• Add or subtract them to get a single resultant constant.

This approach allows us to combine constants efficiently to form part of the simplified expression, resulting in a cleaner and usually easier-to-read format.