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Chapter 0: Problem 48

Evaluate each expression if \(x=-2\) and \(y=3\). \(\frac{2 x-3}{y}\)

Short Answer

Expert verified
\text{-}\frac{\text{7}}{\text{3}}

Step by step solution

01

Understand the Problem

The problem requires evaluating the expression \(\frac{2x-3}{y}\) when \(x = -2\) and \(y = 3\).
02

Substitute Values

Replace \(x\) with \(-2\) and \(y\) with \(3\) in the expression. This gives us: \(\frac{2(-2) - 3}{3}\).
03

Perform Multiplication

Calculate the multiplication step \(2(-2)\), which is \(-4\). The expression now becomes \(\frac{-4 - 3}{3}\).
04

Perform Addition/Subtraction

Simplify the numerator by performing the subtraction \(-4 - 3\), which results in \(-7\). The expression now is \(\frac{-7}{3}\).
05

Perform the Division

Finally, divide \(-7\) by \(3\) to get \(-\frac{7}{3}\).

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Key Concepts

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

Substitution in Expressions
Substitution in algebra involves replacing the variables in an expression with their given values. In this exercise, we are given values for the variables: \(x = -2\) and \(y = 3\). Substituting these values into the expression \(\frac{2x-3}{y}\) means we replace \(x\) with \(-2\) and \(y\) with \(3\). After substitution, the expression becomes \(\frac{2(-2) - 3}{3}\). This step is crucial as it transforms the algebraic expression into a numerical expression that we can easily solve.
Order of Operations
After substitution, it鈥檚 important to follow the order of operations to correctly evaluate the expression. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), guides us on the sequence of steps to perform. In the expression \(\frac{2(-2) - 3}{3}\), we first handle the multiplication inside the parentheses: \(2(-2)\). This gives us \(-4\). Next, we perform the subtraction in the numerator: \(-4 - 3\) which results in \(-7\). Finally, we divide the result by \(3\) to get \(-\frac{7}{3}\).
Rational Numbers
Rational numbers are numbers that can be written as a fraction of two integers, where the denominator is not zero. In this problem, after performing all operations, we get the fraction \(-\frac{7}{3}\). This is a rational number because it is a quotient of two integers (\(-7\) and \(3\)). Understanding rational numbers is important because they include a wide range of values including whole numbers, fractions, and negatives. The simplified form of a rational number is essential for clear and precise mathematical communication.

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Most popular questions from this chapter

Simplify each expression. Assume that all variables are positive when they appear. $$5 \sqrt[3]{2}-2 \sqrt[3]{54}$$

Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive. $$ \frac{\left(4 x^{-1} y^{1 / 3}\right)^{3 / 2}}{(x y)^{3 / 2}} $$

Expressions that occur in calculus are given. Factor each expression. Express your answer so that only positive exponents occur. $$6(6 x+1)^{1 / 3}(4 x-3)^{3 / 2}+6(6 x+1)^{4 / 3}(4 x-3)^{1 / 2} \quad x \geq \frac{3}{4}$$

Simplify each expression. Assume that all variables are positive when they appear. $$\sqrt{9 x^{5}}$$

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{-\sqrt{3}}{\sqrt{5}}$$
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