Which of the following expressions are equivalent to 4-3/4-8? Embark on an intellectual journey as we delve into the intricacies of expression simplification, algebraic techniques, and the profound concept of equivalence. Prepare to witness the transformation of complex expressions into their simplest forms, revealing the underlying mathematical principles that govern their behavior.
In this exploration, we will unravel the mysteries of mathematical equivalence, demonstrating how seemingly different expressions can share the same fundamental value. Through a series of carefully crafted examples, we will illuminate the power of algebraic techniques, such as factoring and substitution, in simplifying expressions and revealing their hidden connections.
Expression Simplification: Which Of The Following Expressions Are Equivalent To 4-3/4-8
The expression “4-3/4-8” can be simplified using the following steps:
- Subtract the numerator from the denominator in the fraction: 4-3/4-8 = 4-1/4-8
- Simplify the fraction: 4-1/4-8 = 3/4-8
- Subtract the second fraction from the first: 3/4-8 = 3/4
- 32/4 =
- 29/4
Therefore, the simplified expression is -29/4.
Algebraic Techniques
One algebraic technique that can be applied to simplify the expression “4-3/4-8” is factoring. Factoring involves finding common factors between the terms in the expression and expressing them as a product.
In this case, we can factor out a 4 from both terms in the expression:
4-3/4-8 = 4(1-3/4)-8
Simplifying further, we get:
4(1-3/4)-8 = 4(1/4)-8 = 1-8 = -7
Therefore, the simplified expression using factoring is -7.
Equivalence of Expressions, Which of the following expressions are equivalent to 4-3/4-8
The simplified expression -29/4 is equivalent to the original expression 4-3/4-8 because they both represent the same value. This means that they can be used interchangeably in mathematical operations without affecting the outcome.
For example, if we have the equation 4-3/4-8 = x, then we can substitute the simplified expression -29/4 for 4-3/4-8 and the equation will still be true:
-29/4 = x
Therefore, the simplified expression -29/4 is equivalent to the original expression 4-3/4-8.
Applications in Mathematics
The simplified expression -29/4 can be used in various applications in mathematics.
- Solving equations:The expression can be used to solve equations involving fractions. For example, if we have the equation 4-3/4-8 = x, we can simplify the expression to -29/4 and then solve for x.
- Solving inequalities:The expression can also be used to solve inequalities involving fractions. For example, if we have the inequality 4-3/4-8< x, we can simplify the expression to -29/4 and then solve the inequality.
- Real-world applications:The expression can be used in various real-world applications, such as calculating the average of a set of numbers or finding the area of a rectangle.
Common Queries
What is the simplified form of 4-3/4-8?
The simplified form is -1.
How can I simplify expressions like 4-3/4-8 using algebraic techniques?
You can use factoring, substitution, or other algebraic techniques to transform the expression into a simpler form.
Why is it important to understand equivalent expressions?
Understanding equivalent expressions allows us to solve equations and inequalities more efficiently and to recognize the underlying mathematical relationships between different expressions.
