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Chapter 1: Problem 233

In the following exercises, evaluate. \(\frac{a+b}{a-b}\) when \(a=-3, b=8\)

Short Answer

Expert verified
-\frac{5}{11}

Step by step solution

01

Substitute the values of a and b

Replace the variables in the expression \(\frac{a+b}{a-b}\) with the given values of \(a = -3\) and \(b = 8\). The expression becomes \(\frac{-3+8}{-3-8}\).
02

Simplify the numerator

Add the values in the numerator: \(-3 + 8 = 5\). So the expression now is \(\frac{5}{-3-8}\).
03

Simplify the denominator

Add the values in the denominator: \(-3 - 8 = -11\). So now, the expression is \(\frac{5}{-11}\).
04

Write the simplified result

The fraction \(\frac{5}{-11}\) can be simplified further to \(-\frac{5}{11}\).

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Key Concepts

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

Substitution of Values
When evaluating expressions with variables, the first step is to substitute the values provided for each variable. For example, in the expression \(\frac{a+b}{a-b}\) given that a = -3 and b = 8, we replace 'a' with -3 and 'b' with 8. This turns the expression into:
\(\frac{-3+8}{-3-8}\).
The goal here is to handle the substitution correctly to proceed smoothly to the next steps. Always double-check the values to ensure they are substituted accurately.
Simplifying Fractions
Simplifying fractions is a crucial part of working with mathematical expressions. Once we substitute the values, we need to simplify both the numerator and the denominator separately first. In our example, the expression \(\frac{-3+8}{-3-8}\) simplifies in the numerator to \(-3 + 8 = 5\) and in the denominator to \(-3 - 8 = -11\).
So, now we have \(\frac{5}{-11}\).
Simplification often involves combining like terms or performing basic arithmetic operations like addition, subtraction, multiplication, or division.
Numerator and Denominator Operations
In fraction expressions, operations in the numerator and the denominator are handled independently. For example, for our expression \(\frac{-3+8}{-3-8}\), let's break this down step by step:
  • First, handle the numerator: \(-3 + 8 = 5\).
  • Next, handle the denominator: \(-3 - 8 = -11\).

This results in \(\frac{5}{-11}\). Always perform these arithmetic operations carefully to simplify correctly.
Order of Operations
Order of operations is fundamental when evaluating any mathematical expression. Ensure you follow this correctly for accurate results:
  • First, handle calculations inside parentheses.
  • Next, conduct multiplication and division from left to right.
  • Lastly, execute addition and subtraction from left to right.

In our example, \(\frac{-3+8}{-3-8}\), addition and subtraction are straightforward since they involve simple terms.
After substitution, evaluate the numerator and denominator separately in order.

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