Problem 233
Question
In the following exercises, evaluate. \(\frac{a+b}{a-b}\) when \(a=-3, b=8\)
Step-by-Step Solution
Verified Answer
-\frac{5}{11}
1Step 1: 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 \(\frac{-3+8}{-3-8}\).
2Step 2: Simplify the numerator
Add the values in the numerator: \(-3 + 8 = 5\). So the expression now is \(\frac{5}{-3-8}\).
3Step 3: Simplify the denominator
Add the values in the denominator: \(-3 - 8 = -11\). So now, the expression is \(\frac{5}{-11}\).
4Step 4: Write the simplified result
The fraction \(\frac{5}{-11}\) can be simplified further to \(-\frac{5}{11}\).
Key Concepts
Substitution of ValuesSimplifying FractionsNumerator and Denominator OperationsOrder of Operations
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.
\(\frac{-3+8}{-3-8}\).
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.
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:
This results in \(\frac{5}{-11}\). Always perform these arithmetic operations carefully to simplify correctly.
- 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:
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.
- 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.
Other exercises in this chapter
Problem 231
In the following exercises, evaluate. \(2 x^{2} y^{3}\) when \(x=-\frac{2}{3}\) and \(y=-\frac{1}{2}\)
View solution Problem 232
In the following exercises, evaluate. \(8 u^{2} v^{3}\) when \(u=-\frac{3}{4}\) and \(v=-\frac{1}{2}\)
View solution Problem 234
In the following exercises, evaluate. \(\frac{r-s}{r+s}\) when $$ r=10, s=-5 $$
View solution Problem 235
Why do you need a common denominator to add or subtract fractions? Explain.
View solution