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Chapter 2: Problem 71

Evaluate the given expression for \(x=2, y=-3,\) and \(z=-4.\) $$y^{2}-3 y+2$$

Short Answer

Expert verified
The value of the expression is 20.

Step by step solution

01

- Substitute the Value of y

Substitute the given value of \( y = -3 \) into the expression \( y^{2} - 3y + 2 \). This will result in the expression: \( (-3)^{2} - 3(-3) + 2 \).
02

- Evaluate the Squared Term

Evaluate the squared term \( (-3)^{2} \). This results in: \( 9 \). Now the expression is: \( 9 - 3(-3) + 2 \).
03

- Multiply and Simplify

Calculate \( -3 \times (-3) \). This results in \( 9 \). Update the expression to: \( 9 + 9 + 2 \).
04

- Add the Constants

Add the constants together: \( 9 + 9 + 2 \) which equals \( 20 \).

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Key Concepts

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

Substitution Method
To evaluate algebraic expressions, we often need to replace variables with their given values. This method is called the substitution method. For instance in our example, we need to replace the variable y with its given value of -3. This transforms the expression from \(y^2 - 3y + 2\) into \((-3)^2 - 3(-3) + 2\). Substitution is crucial because it allows us to convert an abstract algebraic expression into a concrete numerical problem.
  • Start by identifying the variable in the expression.
  • Look up its given value.
  • Replace the variable with this value in the expression.
Using the substitution method helps simplify our work and is a foundational skill in algebra.
Arithmetic Operations
Once we have substituted the values into the expression, we need to perform arithmetic operations such as addition, subtraction, multiplication, and division. In our specific problem, we'll encounter squaring first.
For example, \((-3)^2\) means multiplying -3 by itself, which equals 9.
After squaring, the expression updates to: \(9 - 3(-3) + 2\). Next, we will deal with multiplication:\( -3(-3) \) results in 9. This transforms our expression into \(9 + 9 + 2\).
Remember to carefully follow the order of operations (PEMDAS/BODMAS):
  • Parentheses/Brackets
  • Exponents/Orders
  • Multiplication and Division
  • Addition and Subtraction
Understanding and applying these rules correctly is key to accurately evaluating any algebraic expression.
Simplification Process
The final step in evaluating an algebraic expression is the simplification process. In this phase, we combine like terms and complete any remaining operations to get our final simplified result. In our case, after substituting and performing the arithmetic operations, our expression was reduced to: \( 9 + 9 + 2 \).
Here, we simply add these constants together:
  • Add 9 and 9 to get 18
  • Then add 2 to 18
The final result is 20. Simplification often involves breaking down complex expressions into simpler parts and gradually resolving them. Through practice, the simplification process will become more intuitive and an automatic part of solving algebraic expressions.

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Most popular questions from this chapter

Simplify by combining like terms whenever possible. $$-5 s^{2}+3 s t-s^{2}+6 s^{2}$$

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Evaluate each expression. Round off your answer to the nearest thousandth where necessary. $$(-2.7)^{9}$$
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