Problem 49 - Algebra for College Students

Question

Evaluate the algebraic expressions in Problems 35-57 for the given values of the variables. \(-2 a-3 a+7 b-b, \quad a=-10\) and \(b=9\)

Step-by-Step Solution

Answer

The value of the expression is 104.

Step 1: Substitute the Given Values

Substitute the values of the variables into the expression: \(-2a - 3a + 7b - b\). Given \(a = -10\) and \(b = 9\), the expression becomes \(-2(-10) - 3(-10) + 7(9) - 9\).

Step 2: Simplify the Expression

First, perform the multiplication: \(-2(-10) = 20\) and \(-3(-10) = 30\). So the expression becomes: \(20 + 30 + 7(9) - 9\).

Step 3: Calculate the Values of Each Term

Now, calculate \(7(9) = 63\). The expression is now \(20 + 30 + 63 - 9\).

Step 4: Combine Positive Terms

Add the positive terms together: \(20 + 30 + 63 = 113\).

Step 5: Subtract the Final Term

Finally, subtract \(9\) from \(113\) to get \(113 - 9 = 104\).

Step 6: Conclusion: Evaluate the Expression

The evaluated expression is \(104\).

Key Concepts

Evaluation of Algebraic Expressions

Evaluating algebraic expressions is a fundamental process in algebra that involves finding the value of expressions for given values of the variables. The main goal is to replace each variable with its given numerical value and perform mathematical operations to find a solution. Evaluation is not just about finding an answer; it's about understanding the relationship between variables and numbers.

• First, identify the expression and the values for each variable.
• Substitute these values into the expression.
• Simplify by following the order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction (PEMDAS).
• Make sure every step is logical and correct until you reach the final answer.

Understanding Variables

Variables are symbols, often letters like x, y, and z, used to represent numbers in an equation or expression. They are placeholders that make it possible to form general formulas and equations. In the provided exercise, the variables are "a" and "b."

• Variables allow us to write equations that can handle a broad range of values.
• In algebra, substituting specific values for the variables enables us to calculate a solution to the expression.
• Variables are versatile and help in forming and solving equations in various scientific and mathematical contexts.

Recognizing how variables work and their role is crucial when dealing with algebraic expressions.

Substitution Method

Substitution is the process of replacing variables with given numbers. It is fundamental in evaluating expressions, where you simply "substitute" the variables. In the exercise, when substituting \(a = -10\) and \(b = 9\), each instance of "a" and "b" in the expression is replaced with these values.

• Make sure to substitute each variable correctly and consistently throughout the expression.
• Handle negative numbers meticulously to avoid errors during calculation.
• Check your substitutions to ensure that none of the initial values are overlooked or misplaced.

Substitution is key for getting the right answers and is a skill that needs practice to master fully.

Simplification of Algebraic Expressions

Simplification is about reducing an expression to its simplest form. After substitution, simplification involves performing arithmetic operations so the expression can easily yield a clear answer.
First, calculate each term separately, handling any negative numbers with care, as done in \(-2(-10)\) which becomes 20. Then:

• Add or subtract terms step-by-step, while being cautious with the signs of the numbers involved.
• Combine like terms to make calculations simpler and more efficient.
• Lastly, ensure that the final expression is reduced, leaving no further simplifications or operations needed.

By simplifying accurately, the complexity of an expression is decreased, making it easier to understand and evaluate.