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Chapter 1: Problem 87

Evaluate each expression for \(x=6, y=-4,\) and \(a=3\) $$ (-5+x)(-3+y)(3-a) $$

Short Answer

Expert verified
The value of the expression is 0.

Step by step solution

01

- Substitute the values

First, substitute the given values of the variables into the expression. Given are: \(x=6\), \(y=-4\), and \(a=3\). Thus, the expression \((-5+x)(-3+y)(3-a)\) becomes \((-5+6)(-3-4)(3-3)\).
02

- Simplify inside the parentheses

Now, simplify inside each parentheses: \((-5+6)=1\), \((-3-4)=-7\), and \((3-3)=0\). Thus, the expression is now \((1)(-7)(0)\).
03

- Multiply the simplified terms

Finally, multiply the simplified terms together: \((1)\cdot(-7)\cdot(0)=0\). Hence, the value of the expression is 0.

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Key Concepts

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

substitution
Substitution is the process of replacing variables with their specific values. In our exercise, we start with the expression (-5+x)(-3+y)(3-a) and substitute the given values for each variable. Given: x=6, y=-4, and a=3. Thus, when we substitute these values into the expression, it becomes (-5+6)(-3-4)(3-3). This step transforms the algebraic expression into a numerically calculable form.Explanation points:
  • Identify the given values for each variable.
  • Replace each variable in the expression with its corresponding value.
  • Write down the new expression with all numerical values.
Substitution helps in converting an abstract algebraic problem into a simpler arithmetic problem.
simplifying expressions
Simplifying expressions involves performing basic arithmetic operations, such as addition and subtraction, within the parentheses. After substitution, our expression becomes (-5+6)(-3-4)(3-3). Let's simplify it step by step. The first parenthesis simplifies to 1: (-5+6)=1. The second parenthesis simplifies to -7: (-3-4)=-7. The third parenthesis simplifies to 0: (3-3)=0. Now, we have transformed the expression to (1)(-7)(0).Explanation points:
  • Focus on one parenthesis at a time.
  • Perform the operations inside each parenthesis.
  • Write down the simplified form of each part of the expression.
By simplifying expressions, we break down complex problems into smaller, more manageable parts.
multiplication of integers
Multiplication of integers is the final step in evaluating the simplified expression. After substitution and simplification, we have the expression (1)(-7)(0). When multiplying integers, there are simple rules we need to follow:
  • Any number multiplied by 0 is 0.
  • The product of a positive and a negative number is negative.
  • The product of two positive or two negative numbers is positive.
In our case, multiplying (1) and (-7) results in (-7), but when we multiply (-7) by (0), we get 0. Thus, the final result of (1)(-7)(0) is 0.Explanation points:
  • Apply multiplication rules systematically.
  • Understand that zeros nullify the product.
  • Write the final answer after calculations.
The multiplication of integers ties everything together to deliver the final evaluated number.

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