Problem 98 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

0

Step 1: 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

Step 2: 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\(1)(-7)(0)\)

Step 3: 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: \(0\)

Key Concepts

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.