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Chapter 1: Problem 91

Evaluate each expression for \(x=6, y=-4,\) and \(a=3\) $$ \frac{2 y^{2}-x}{a+10} $$

Short Answer

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Step by step solution

01

- Substitute the values

Replace the variables in the expression with the given values: \( x = 6 \), \( y = -4 \), and \( a = 3 \). The expression becomes: \[ \frac{2 (-4)^{2} - 6}{3 + 10} \]
02

- Evaluate the numerator

First, calculate the exponent and multiplication inside the numerator: \[ (-4)^{2} = 16 \] Then multiply by 2: \[ 2 \times 16 = 32 \] Subtract the value of \( x \) from this product: \[ 32 - 6 = 26 \]
03

- Evaluate the denominator

Add the values in the denominator: \[ 3 + 10 = 13 \]
04

- Simplify the fraction

Divide the evaluated numerator by the evaluated denominator: \[ \frac{26}{13} = 2 \]

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Key Concepts

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

Substitution in Algebra
Substitution is a method used to solve mathematical expressions by replacing variables with their given values. This approach simplifies the expression so you can evaluate it step by step. Let's consider the provided exercise:
For the given expression \(\frac{2 y^{2}-x}{a+10}\) and values \(x = 6\), \(y = -4\), and \(a = 3\).
Using substitution, we replace \(x\) with 6, \(y\) with -4, and \(a\) with 3. Thus our expression becomes:
\(\frac{2 (-4)^{2} - 6}{3 + 10}\).
This initial step ensures that we are working with known quantities and prepares the expression for further simplification.
Order of Operations
The order of operations is a set of rules that dictate the correct sequence to evaluate a mathematical expression. The common acronym PEMDAS helps remember the order:
  • P: Parentheses first
  • E: Exponents (i.e., powers and roots, etc.)
  • M/D: Multiplication and Division (left to right)
  • A/S: Addition and Subtraction (left to right)

In our example, \(\frac{2 (-4)^{2} - 6}{3 + 10}\), we follow these rules:
  • First, solve the exponent: \((-4)^{2} = 16\)
  • Next, perform the multiplication: \2 \times 16 = 32\
  • Then, handle the subtraction in the numerator: \32 - 6 = 26\
  • Finally, add the values in the denominator: \3 + 10 = 13\

We end up with \(\frac{26}{13}\)
Simplifying Fractions
Simplifying fractions means expressing a fraction in its simplest form. This involves dividing the numerator and the denominator by their greatest common divisor (GCD).
In our example \(\frac{26}{13}\), the numerator (26) and the denominator (13) share a common divisor, which is 13.
Performing the division: \(\frac{26 \div 13}{13 \div 13} = \frac{2}{1}\).
This leaves us with a simplified result of 2.
Simplifying fractions helps us obtain a more straightforward, clean result, making it easier to understand and work with.

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