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Chapter 2: Problem 70

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

Short Answer

Expert verified
-75

Step by step solution

01

Substitute the Given Values

Substitute the values of the variables into the expression. Given: \(x = 2, y = -3, z = -4\). The expression is \((y - 2z)(y + 3z)\). Substitute \(y = -3\) and \(z = -4\) into the expression: \((-3 - 2(-4))(-3 + 3(-4))\).
02

Simplify Inside the Parentheses

Simplify the expressions inside the parentheses: Find the value of \(-3 - 2(-4)\). Calculate \(-2(-4) = 8\), then \(-3 + 8 = 5\).Find the value of \(-3 + 3(-4)\). Calculate \(3(-4) = -12\), then \(-3 - 12 = -15\).
03

Multiply the Simplified Expressions

Multiply the simplified expressions: \((5)(-15) = -75\).

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Key Concepts

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

substitution method
The substitution method involves replacing the variables in an algebraic expression with given values. In this exercise, we are given specific values for three variables: \(x = 2\), \(y = -3\), and \(z = -4\). Our goal is to substitute these values into the given expression \((y - 2z)(y + 3z)\). This method helps turn the general expression into a numerical one, which we can then simplify and evaluate.

Here are the steps:
  • Identify the variables in the expression.
  • Replace each variable with the given number.
  • Rewrite the expression with these substitutions, turning it from variable form into numeric form.
Applying the substitution method to our expression, we substitute \(y = -3\) and \(z = -4\):

\((-3 - 2(-4))(-3 + 3(-4))\). Now we need to simplify this further.
simplifying algebraic expressions
Simplifying algebraic expressions means reducing them into their simplest form. After substituting the numbers into the expression, we encounter nested operations that must be simplified step-by-step.

In our case:
  • First, look inside the parentheses.
  • Perform any multiplications or divisions.
  • Then do any additions or subtractions.
Let's simplify the inner expressions:
  • For \(-3 - 2(-4)\): Compute \(-2(-4) = 8\), then add to \(-3\) giving \(-3 + 8 = 5\).
  • For \(-3 + 3(-4)\): Compute \(3(-4) = -12\), then add to \(-3\) giving \(-3 - 12 = -15\).
Our expression now simplifies to \((5)(-15)\).
multiplication of integers
Multiplying integers is straightforward but remembering the sign rules is crucial for the correct answer. The sign rules are:
  • Multiplying two positive numbers results in a positive number.
  • Multiplying two negative numbers also results in a positive number.
  • Multiplying a positive number by a negative number results in a negative number.
  • Likewise, multiplying a negative number by a positive number results in a negative number.
In our simplified expression, we multiply \(5\) by \(-15\):
  • Since one number is positive and the other is negative, the result is negative.
  • Performing the multiplication, we get \(5 \cdot (-15) = -75\).
Therefore, the final evaluated expression is \(-75\).

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Most popular questions from this chapter

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

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