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Chapter 1: Problem 13

Evaluate each expression when \(x=-2, y=7,\) and \(z=-3\) $$ x^{2}+x y+10 $$

Short Answer

Expert verified
The evaluated expression when \(x=-2, y=7,\) and \(z=-3\) is \(0\).

Step by step solution

01

Plug in the given values of x, y, and z

Replace x, and y with the given values -2 and 7 respectively: \[ (-2)^2+(-2)(7)+10 \]
02

Compute the squares and multiplications

Compute the square of -2 and the multiplication -2 times 7: \[ (4)+(-14)+10 \]
03

Compute the final result

Add all the numbers together to get the final result: \[ 4 - 14 + 10 = 0 \] Therefore, the evaluated expression when \(x=-2, y=7,\) and \(z=-3\) is \(0\).

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Key Concepts

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

Substitution in Expressions
Substitution in expressions is a fundamental concept in algebra. It involves replacing variables in an expression with known values to simplify or evaluate the expression.
In our example, we substitute the variables \(x\) and \(y\) with the numbers \(-2\) and \(7\) respectively. This turns an abstract algebraic expression like \(x^2 + xy + 10\) into a simple arithmetic problem.

Steps to follow when substituting:
  • Identify the variables in the expression and the values that correspond to them.
  • Carefully replace each variable in the expression with its given value.
  • Ensure that substitution is done consistently to avoid errors.
Substitution helps convert a symbolic expression into a numerical one, making complex problems easier to solve.
Integer Arithmetic
Integer arithmetic involves performing mathematical operations with whole numbers, which include positive numbers, negative numbers, and zero.
After substitution, we need to carry out integer arithmetic operations on the numbers \((-2)^2\), \(-2 \times 7\), and \(10\).

Key operations and rules:
  • Addition/Subtraction: When adding or subtracting integers, pay attention to the signs. For instance, \(-14\) is added to 4 and 10 in the example, requiring careful computation of signs.
  • Multiplication: Multiplying a negative number by a positive number results in a negative product. In this exercise, \(-2 \times 7\) equals \(-14\).
  • Squaring: Squaring a negative number results in a positive value, as shown by \((-2)^2 = 4\).
Mastering integer arithmetic is crucial as it forms the basis for solving a wide range of mathematical problems.
Evaluating Expressions
Evaluating expressions means computing their value after substituting variables and performing arithmetic operations. It is the process of transforming an expression into a single numerical result.
Let's break down the evaluation in the example:
  • First, substitute the given values for each variable in the expression \(x^2 + xy + 10\).
  • Next, perform any necessary arithmetic operations, such as multiplication and addition, ensuring to maintain the order of operations (often remembered by PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).
  • Finally, combine all terms to reach the final value, ensuring no steps are missed. In this example, the transformation \(4 - 14 + 10\) simplifies to 0.
Evaluating expressions accurately allows you to solve and understand complex algebraic problems with ease.

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Most popular questions from this chapter

Divide. $$\frac{-84}{-210}$$

Use the order of operations to simplify. $$(9-14)^{2}-(-3)(6)$$

Rewrite each expression using the distributive property. Simplify if possible. $$2(-6+5+3)$$

Write a mathematical expression for each and simplify. \(-25\) less than \(-19\).

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