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Chapter 1: Problem 98

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

Short Answer

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Step by step solution

01

Substitute the values

Replace each variable in the expression with the given values: \(x=6\), \(y=-4\), and \(a=3\). The expression becomes ewline ewline ewline\begin{align*}(-5+6)(-3+(-4))(3-3)ewlineewline ewlineewline eend{align*}imewline
02

Simplify each term inside the parentheses

Calculate each term inside the parentheses: ewline \(-5+6=1\) ewline \-3+(-4)=-7\ ewline Thus, the expression simplifies to: ewline ewline\(1)(-7)(0)\)
03

Multiply the terms

Finally, multiply the simplified terms together: ewlineiintMultiplying any number by 0 results in: \(1 \times -7 \times 0 = 0\) ewline Thus, the result 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 first and one of the most crucial steps in evaluating an algebraic expression. Here, we replace the variables in the expression with the given numerical values. In our exercise, the original expression is \((-5 + x)(-3 + y)(3 - a)\).

We are given the values for \(x\), \(y\), and \(a\): \x = 6\, \y = -4\, and \a = 3\.

By substituting these values into the expression, we replace \(x\) with \(6\), \(y\) with \(-4\), and \(a\) with \(3\). The expression then becomes:

\( (-5 + 6)(-3 + (-4))(3 - 3)\).

This transformed version is now ready for further simplification. Understanding substitution is key to solving algebraic expressions accurately.
Simplification
Simplification is the process of reducing expressions to their most basic form. With our substituted expression \((-5 + 6)(-3 + (-4))(3 - 3)\), we need to simplify each of the terms inside the parentheses.

Let's take it step by step:
  • First term: \(-5 + 6 = 1\)

  • Second term: \(-3 + (-4) = -7\)

  • Third term: \(3 - 3 = 0\)

The expression simplifies to \(1 \times (-7) \times 0\).

Simplifying terms inside the parentheses helps to make multiplication much easier and ensures we achieve the correct final result.
Multiplication
After substitution and simplification, we arrive at a simple expression that's ready to be multiplied: \(1 \times (-7) \times 0\).

It's important to remember the rules of multiplication, especially when dealing with negative numbers and zero.
  • Multiplying any number by zero results in zero. This is a fundamental property in mathematics.

  • Whether the other terms are positive or negative doesn't matter if one of them is zero.

Consequently, the result of \(1 \times -7 \times 0\) is \0\.

Understanding the multiplication rules, especially with zero, is essential for solving algebraic expressions accurately.

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Most popular questions from this chapter

Evaluate each expression for \(x=6, y=-4,\) and \(a=3\) \(5 x^{2}-4 y^{2}\)

Use the distributive property to rewrite each expression. $$ -\frac{4}{3}(12 y+15 z) $$

Evaluate each expression for \(x=6, y=-4,\) and \(a=3\) \(\left(\frac{5}{6} x+\frac{3}{2} y\right)\left(-\frac{1}{3} a\right)\)

Use the distributive property to rewrite each expression. $$ -\frac{1}{5}(5 x-7) $$

To find the average (mean) of a group of numbers, we add the numbers and then divide the sum by the number of terms added. Work Exercises 139-142 in order, to find the average of \(23,18,13,-4,\) and -8 Find the sum of the given group of numbers.
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