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Chapter 3: Problem 43

Explain how to evaluate an expression. $$2 x-4 y ; x=3, y=-2$$

Short Answer

Expert verified
Answer: The value of the expression when \(x=3\) and \(y=-2\) is 14.

Step by step solution

01

Write down the given expression.

We have the expression: $$2x - 4y$$ And the given values: $$x=3, \ y=-2$$
02

Substitute the values of x and y into the expression.

Replace \(x\) with \(3\), and \(y\) with \(-2\) in the expression: $$2(3) - 4(-2)$$
03

Begin simplifying using the order of operations.

First, perform the multiplication: $$6 + 8$$
04

Complete the simplification by performing the addition.

Add the two numbers together: $$6 + 8 = 14$$ The evaluated expression is $$14$$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution Method
The substitution method is a key tool in mathematics used to evaluate expressions or solve equations. It's like a puzzle that involves replacing variables with given values.
  • Identify the variables: Before substituting, identify all variables in the expression. For instance, in the expression \(2x - 4y\), we have two variables, \(x\) and \(y\).
  • Gather the needed values: Make sure to have the values for the variables at hand. In this case, we know \(x = 3\) and \(y = -2\).
  • Perform the substitution: Replace each variable with its corresponding value. This transforms our expression into \(2(3) - 4(-2)\). Notice how each variable is swapped out for the number given.
Substitution helps to simplify the expression so that further calculations become straightforward. It鈥檚 a simple yet powerful method that forms the backbone of many mathematical operations.
Order of Operations
The order of operations, often recalled by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)), guides us on the correct sequence to evaluate an expression.
  • Start with Multiplication/Division: In our substituted expression \(2(3) - 4(-2)\), we first perform the multiplication \(2 \times 3 = 6\) and \(-4 \times -2 = 8\).
  • Then, do Addition/Subtraction: Be careful as the order matters. Once the multiplications are done, what remains is \(6 + 8\).
By following the order of operations precisely, similar to following a recipe, you prevent common mistakes and ensure a correct answer every time!
Simplifying Expressions
Simplifying expressions is like cleaning your workspace before starting a new project. It makes everything clearer and easier to handle. Once you've done substitutions and respected the order of operations, simplifying is the last step to find your answer.
  • Combine like terms: This involves reducing the expression as much as possible, often through addition or subtraction. In our example, after substituting, we simplified \(6 + 8\) directly because these are both constants.
  • Focus on clarity: Ensure every step follows logically from the last. Each operation narrows down the expression until you reach the final and simplest form, which here is \(14\).
Simplifying expressions is crucial because it's the final step that provides clarity and resolution. It transforms a complex expression into a final, easy-to-read number or simpler form.

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