Problem 56 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression. $$-10 b+13+(-b)+(-4)$$

Step-by-Step Solution

Answer

The simplified form of the expression is \(-11b+ 9\).

Step 1: Identify like terms

In the given expression \(-10b+13+(-b)+(-4)\), like terms are ones that have the same variable to the same power. Here the like terms are \(-10b\) and \(-b\) (these are both 'b' terms), and 13 and \(-4\) (these are constants without variables).

Step 2: Combine like terms

Next, combine the like terms by adding or subtracting them. To do this, add the coefficients (the numbers in front of the variable) together. So \(-10b+ -b = -11b\). Not forgetting to add the constant values: \(13+ -4 = 9\).

Step 3: Write out the simplified expression

The final step is to write down the simplified expression by adding together the results from Step 2. That gives us \(-11b+ 9\). This is the simplified form of the original expression.

Key Concepts

Combining Like Terms

Combining like terms is a fundamental skill in algebra that makes simplifying expressions easier. Like terms are terms that share the same variables raised to the same powers. In the expression

• \(-10b + 13 + (-b) + (-4)\),

terms with the variable \(b\) are like terms, specifically \(-10b\) and \(-b\). Similarly, constant terms like 13 and \(-4\) are also considered like terms, as they don’t contain variables.

To combine, simply add or subtract their coefficients. Remember that a coefficient is the number in front of a variable. For the given expression: the coefficients for \(b\) terms are -10 and -1 (implied in \(-b\)). Adding these gives \(-11b\). For the constants, 13 and -4 add up to 9. Combining like terms tidies the expression, leaving you with a clearer, more manageable equation. In this case, the simplified expression is \(-11b + 9\). Each step connects to making further algebraic manipulations easier.

Algebraic Expressions

Algebraic expressions are mathematical statements that include variables, coefficients, and constants. They can involve various operations such as addition, subtraction, multiplication, and sometimes division. The key component is the variable, which can represent numbers within any calculations.

In our example of an algebraic expression, we have:

• The variable is \(b\), which can take any value.
• The coefficients are -10 and -1, which are the numbers multiplied by \(b\).
• Lastly, the constants are 13 and -4, which do not depend on any variable.

Understanding how these pieces fit together helps in manipulating them in steps like simplifying. An expression like \(-10b + 13 + (-b) + (-4)\) simply sets the stage for solving or simplifying even more complex expressions.

Grasping the roles of each part in algebraic expressions is essential for engaging with higher-level math problems.

Polynomial Simplification

Polynomial simplification involves reducing a polynomial, a type of algebraic expression, to its simplest form. This process makes the polynomial easier to handle and analyze. Polynomials consist of a sum of terms, each term being a product of a coefficient and a variable that is raised to a non-negative integer power.

With the expression \(-10b + 13 + (-b) + (-4)\), simplifying involves:

• Identifying groups of like terms.
• Performing arithmetic within these groups—namely addition and subtraction.
• Rewriting the polynomial with combined terms.

By simplifying, the expression transforms into \(-11b + 9\), presenting an easier form without unnecessary complexity.

Simplifying not only reduces clutter but also prepares an expression for evaluating at given values, differentiating, integrating, or finding solutions to equations. Recognize this process as essential for making algebraic problem-solving efficient and clear.