Problem 10
Question
For exercises 1-66, simplify. $$ \frac{3 x-12}{15} $$
Step-by-Step Solution
Verified Answer
The simplified form is \( \frac{x-4}{5} \).
1Step 1: Factor the Numerator
First, factor out the greatest common factor (GCF) from the numerator. The GCF of the terms in the numerator, 3 and -12, is 3. Factoring this out, the numerator becomes: \(3(x - 4)\).
2Step 2: Simplify the Fraction
Now that the numerator is factored, the expression looks like this: \(\frac{3(x - 4)}{15}\). Next, simplify the fraction by dividing both the numerator and the denominator by their GCF, which is 3.
3Step 3: Divide by the GCF
Divide both the numerator and the denominator by 3: \(\frac{3(x - 4)}{15} = \frac{3(x - 4)}{3 * 5} = \frac{x - 4}{5}\).
Key Concepts
Greatest Common FactorFactoring ExpressionsSimplifying Fractions
Greatest Common Factor
The Greatest Common Factor (GCF) of two or more numbers is the largest number that divides each of them without leaving a remainder. In our example, we need to find the GCF of 3 and -12. The GCF is 3 because 3 is the largest number that divides both 3 and -12 evenly.
To factor out the GCF, we divide each term in the expression by the GCF and write the expression as a product of the GCF and another factor. For \(3x-12\), factoring out 3 gives us \(3(x-4)\).
Understanding the GCF is essential because it helps us simplify complex algebraic expressions, making them easier to work with and solve.
To factor out the GCF, we divide each term in the expression by the GCF and write the expression as a product of the GCF and another factor. For \(3x-12\), factoring out 3 gives us \(3(x-4)\).
Understanding the GCF is essential because it helps us simplify complex algebraic expressions, making them easier to work with and solve.
Factoring Expressions
Factoring involves writing an expression as a product of its factors. Factors are numbers or expressions that multiply together to produce a given number or expression.
For algebraic expressions, we often start by factoring out the GCF. In our example, the expression \(3x-12\) can be factored by taking out the GCF of 3, resulting in \(3(x-4)\).
Factoring is a crucial step in simplifying algebraic expressions because it allows us to break down a complex expression into simpler components. This is particularly useful in solving equations, simplifying fractions, and performing other algebraic operations.
For algebraic expressions, we often start by factoring out the GCF. In our example, the expression \(3x-12\) can be factored by taking out the GCF of 3, resulting in \(3(x-4)\).
Factoring is a crucial step in simplifying algebraic expressions because it allows us to break down a complex expression into simpler components. This is particularly useful in solving equations, simplifying fractions, and performing other algebraic operations.
Simplifying Fractions
Simplifying fractions means reducing them to their simplest form. This involves dividing the numerator and the denominator by their GCF.
For example, after factoring the numerator in \( \frac{3(x-4)}{15} \), we can simplify the fraction by dividing both the numerator and the denominator by their GCF, which is 3. This gives us \( \frac{3(x-4)}{3 \times 5} \). Dividing both the top and bottom by 3, we simplify the fraction to \ \frac{x-4}{5} \.
Simplifying fractions is an essential skill for making complex fractions more manageable and for solving many types of math problems more easily.
For example, after factoring the numerator in \( \frac{3(x-4)}{15} \), we can simplify the fraction by dividing both the numerator and the denominator by their GCF, which is 3. This gives us \( \frac{3(x-4)}{3 \times 5} \). Dividing both the top and bottom by 3, we simplify the fraction to \ \frac{x-4}{5} \.
Simplifying fractions is an essential skill for making complex fractions more manageable and for solving many types of math problems more easily.
Other exercises in this chapter
Problem 10
For exercises 1-12, use prime factorization to find the least common denominator. $$ \frac{1}{40 x y^{2} z} ; \frac{1}{42 x y z^{3}} $$
View solution Problem 10
For exercises 7-32, simplify. $$ \left(\frac{8 p-24}{9 p+18}\right)\left(\frac{27}{32}\right) $$
View solution Problem 11
The relationship of \(x\) and \(y\) is an inverse variation. When \(x=2, y=6\). a. Find the constant of proportionality, \(k\). b. Write an equation that repres
View solution Problem 11
For exercises \(9-24\), evaluate or simplify. $$ \frac{\frac{5 x}{4}}{\frac{2 x^{2}}{15}} $$
View solution