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Chapter 2: Problem 67

Evaluate each expression for \(x=-2, y=4,\) and \(z=-1 .\) $$ \frac{5 y}{z} $$

Short Answer

Expert verified
The expression evaluates to -20.

Step by step solution

01

Substitute the Given Values

Begin by substituting the given values of the variables into the expression. We have \( y = 4 \) and \( z = -1 \), so substitute these values into \( \frac{5y}{z} \). The expression becomes \( \frac{5(4)}{-1} \).
02

Simplify the Numerator

Next, simplify the expression in the numerator by multiplying the constant with \( y \). Calculate \( 5 \times 4 = 20 \). Now the expression simplifies to \( \frac{20}{-1} \).
03

Simplify the Fraction

Finally, divide the numerator by the denominator. Thus, \( \frac{20}{-1} = -20 \). This gives us the final value of the expression.

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Key Concepts

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

Substitution in Algebra
Substitution in algebra is a key tool that allows you to solve expressions by replacing variables with given values. Imagine an expression like \( \frac{5y}{z} \) where the variables \( y \) and \( z \) are placeholders for numbers. In this exercise, \( y = 4 \) and \( z = -1 \). By substituting these values into the expression, you transform it into arithmetic that you can solve easily.
Substitution steps:
  • Identify the variables in the expression.
  • Replace each variable with the given numerical value.
  • Simplify the resulting expression to find the value.
The goal of substitution is to convert an abstract algebraic expression into a concrete arithmetic problem that you can compute with real numbers. It simplifies the process and helps in understanding variable dynamics.
Simplifying Fractions
Fractions, at their core, are a way of expressing division but in a different form. Simplifying fractions means reducing them to their simplest form without changing their value. In the provided expression, after substitution, you are left with the fraction \( \frac{20}{-1} \). This fraction signifies that 20 is divided by -1. Simplifying fractions involves several steps:
  • Multiply or divide the numerator (top number) and the denominator (bottom number) by the same non-zero number if possible.
  • Sometimes, fractions can be simplified by canceling common factors from the numerator and the denominator.
  • Be cautious about the sign of numbers, especially when it is negative as simplification should retain the sign.
In this case, dividing 20 by -1 simply gives you -20 because dividing by -1 is the same as multiplying by -1, flipping the sign of the number.
Negative Numbers in Math
Understanding negative numbers is essential in math, especially when working with expressions. A negative number simply indicates a direction on the number line opposite to positive numbers. Being comfortable with them can make a big difference in calculations.
  • Negative numbers appear when you subtract a larger number from a smaller one or work with values less than zero.
  • When multiplying or dividing by a negative number, the sign of the result changes. For example, \( 5 \times -1 = -5 \).
  • Combining two negatives ends up as a positive, for example, \( -2 \times -3 = 6 \).
In this particular expression \( \frac{20}{-1} \), the presence of a negative number in the denominator results in the entire expression yielding a negative outcome. Therefore, understanding the rules for negatives helps anticipate the results in algebraic expressions.

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