Problem 29
Question
Add or subtract as indicated. Be sure to express your answers in simplest forn. (Objective 1) $$\frac{2(3 x-4)}{7 x^{2}}-\frac{7 x-8}{7 x^{2}}$$
Step-by-Step Solution
Verified Answer
The answer is \(-\frac{1}{7x}\).
1Step 1: Combine Fractions
The given problem is \(\frac{2(3x-4)}{7x^2} - \frac{7x-8}{7x^2}\). Both expressions have the same denominator, \(7x^2\). Thus, we can combine the numerators directly: \(2(3x-4) - (7x-8)\).
2Step 2: Distribute and Simplify
Expand \(2(3x-4)\) to get \(6x - 8\). Now, calculate the new numerator: \(6x - 8 - (7x - 8)\). Distribute the negative sign to both terms in \(7x - 8\), resulting in \(6x - 8 - 7x + 8\).
3Step 3: Simplify Expression
Combine the like terms in the numerator: \(6x - 7x = -x\) and \(-8 + 8 = 0\). The simplified numerator is \(-x\).
4Step 4: Write Final Expression
The simplified fraction is \(\frac{-x}{7x^2}\). Further simplification involves dividing both the numerator and the denominator by \(x\), provided that \(x eq 0\), leading to \(-\frac{1}{7x}\).
Key Concepts
Algebraic ExpressionsSimplifying FractionsLike Terms
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They form the foundation of algebra, allowing for the modeling and solving of various mathematical problems. In algebraic expressions, coefficients are numbers multiplied by variables. Variables are symbols representing numbers whose specific values are often unknown. For instance, in the expression \(2(3x-4)\), the term \(3x\) includes the coefficient 3 and the variable \(x\). Meanwhile, \(-4\) is a constant term.
Understanding algebraic expressions involves recognizing these components and how they interact. When working with such expressions, you'll frequently encounter operations like addition, subtraction, and multiplication, often within the context of solving equations or simplifying expressions.
Understanding algebraic expressions involves recognizing these components and how they interact. When working with such expressions, you'll frequently encounter operations like addition, subtraction, and multiplication, often within the context of solving equations or simplifying expressions.
Simplifying Fractions
Simplifying fractions involves reducing them to their simplest form, making calculations easier and results neater. This process includes eliminating any common factors from the numerator and the denominator.
In the given exercise, after combining and simplifying the numerator of the fraction, we reached a point where the simplified expression was \(-\frac{x}{7x^2}\). To simplify further, notice that both the numerator \(-x\) and the denominator \(7x^2\) include the variable \(x\) as a factor.
To simplify, divide both the numerator and the denominator by \(x\), assuming \(x eq 0\). This results in \(-\frac{1}{7x}\). By removing common factors, you ensure the expression is in its simplest form, which is crucial for clarity and ease of computation.
In the given exercise, after combining and simplifying the numerator of the fraction, we reached a point where the simplified expression was \(-\frac{x}{7x^2}\). To simplify further, notice that both the numerator \(-x\) and the denominator \(7x^2\) include the variable \(x\) as a factor.
To simplify, divide both the numerator and the denominator by \(x\), assuming \(x eq 0\). This results in \(-\frac{1}{7x}\). By removing common factors, you ensure the expression is in its simplest form, which is crucial for clarity and ease of computation.
Like Terms
Like terms are terms in an expression that have identical variable parts. These terms can and should be combined to simplify expressions and make them more manageable. For example, in algebraic expressions, \(6x\) and \(-7x\) are like terms because they both involve the variable \(x\).
In our example, after expanding and expecting both fractions' numerators, we dealt with \(6x - 7x\) and \(-8 + 8\). The like terms \(6x\) and \(-7x\) combine to form \(-x\), and the constants \(-8\) and \(+8\) cancel each other out to zero.
Understanding and identifying like terms is essential as it allows us to combine them easily, facilitating the simplification process and thus leading us to a simplified form of the expression. The concept of like terms is fundamental in both algebraic operations and in solving equations.
In our example, after expanding and expecting both fractions' numerators, we dealt with \(6x - 7x\) and \(-8 + 8\). The like terms \(6x\) and \(-7x\) combine to form \(-x\), and the constants \(-8\) and \(+8\) cancel each other out to zero.
Understanding and identifying like terms is essential as it allows us to combine them easily, facilitating the simplification process and thus leading us to a simplified form of the expression. The concept of like terms is fundamental in both algebraic operations and in solving equations.
Other exercises in this chapter
Problem 28
\(\frac{n}{57-n}=10+\frac{2}{57-n}\)
View solution Problem 29
Perform the indicated multiplications and divisions and express your answers in simplest form. $$\frac{2 x^{2}-3 x y+y^{2}}{4 x^{2} y} \div \frac{x^{2}-y^{2}}{6
View solution Problem 29
Simplify each algebraic fraction. $$\frac{6 x^{3}-15 x^{2} y}{6 x^{2}+24 x y}$$
View solution Problem 29
For Problems 1-40, perform the indicated operations and express answers in simplest form. $$ \frac{a}{a b+b^{2}}-\frac{b}{a^{2}+a b} $$
View solution