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Chapter 2: Problem 13

Evaluate each expression when \(x=-3, y=-5,\) and \(z=0.\) $$\frac{x-y^{2}}{2 y-1+x}$$

Short Answer

Expert verified
The value of the expression evaluated at \(x=-3, y=-5,\) and \(z=0\) is 2.

Step by step solution

01

Substitute Variables

Substitute \(x=-3, y=-5,\) and \(z=0\) into the original expression: \[\frac{-3-(-5)^{2}}{2(-5)-1+(-3)}\]
02

Calculate Power

Remembering that order of operations always requires you to calculate powers before doing other calculations, find the square of \(-5\) which is \(25\) within the expression: \[\frac{-3-25}{2(-5)-1+(-3)}\]
03

Simplify Numerator

Now, further simplify the numerator by placing values: \[-3-25=-28\] The equation now becomes: \[\frac{-28}{2(-5)-1+(-3)}\]
04

Simplify Denominator

Simplify the denominator: \[2(-5)-1+(-3) = -10-1-3 =-14\] Put this value in the denominator in the equation: \[\frac{-28}{-14}\]
05

Calculate the Fraction

Lastly, put the \(x\) and \(y\) values in the equation and solve it: \[\frac{-28}{-14} = 2\]

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Key Concepts

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

Order of Operations
Order of operations is a fundamental concept in algebra, ensuring that everyone solves expressions in the same way. When tackling expressions, always follow the sequence known as PEMDAS or BIDMAS, which stands for:
  • Parentheses/Brackets
  • Exponents/Indices
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)
In our problem, the expression \[\frac{-3-(-5)^{2}}{2(-5)-1+(-3)}\] involves several operations. Start with exponents: calculate \((-5)^2 = 25\).
Then move to multiplication in the denominator: calculate \(2(-5) = -10\). After solving these, proceed with the rest of the operations according to the order.
Substitution in Expressions
Substitution involves replacing variables in an expression with given values. This process simplifies solving the problem by converting the expression into numbers only. In this exercise:
  • Replace \(x\) with \(-3\)
  • Replace \(y\) with \(-5\)
  • Replace \(z\) with \(0\)
The expression becomes:\[\frac{-3-(-5)^{2}}{2(-5)-1+(-3)}\]Substituting allows us to focus solely on arithmetic operations, making calculation straightforward.
Simplifying Fractions
Simplifying fractions means reducing them to their simplest form. This happens by dividing both the numerator and the denominator by their greatest common factor. In the exercise, after substitution and simplification of operation steps, we arrived at:\[\frac{-28}{-14}\]Recognize the negatives cancel each other, giving us \(2\).
This simplification shows understanding of how fractions work: \(-28\div-14\) results in \(2\). Always check to reduce fractions to simplify calculations.

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