Problem 104 - College Algebra

Question

Determine whether each statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{2 x-1}{x-7}+\frac{3 x-1}{x-7}-\frac{5 x-2}{x-7}-0$$

Step-by-Step Solution

Answer

The statement \( \frac{2 x-1}{x-7} + \frac{3 x-1}{x-7} - \frac{5 x-2}{x-7} = 0 \) is true provided \( x \neq 7 \)

Step 1: Identification of Like Terms

The terms \(\frac{2 x-1}{x-7}, \frac{3 x-1}{x-7}, \frac{5 x-2}{x-7}\) have the same denominator \(x - 7\). Hence, they are like terms.

Step 2: Combining Like Terms

Since like terms can be added together and subtracted, combine these like terms to simplify the expression to \(\frac{2 x-1 +3 x-1 -5 x+2 }{x-7}\)

Step 3: Simplify Numerator

Simplify the numerator by performing the required addition and subtraction operations, this will result in \(\frac{0}{x-7}\)

Step 4: Simplify Entire Expression

The expression \(\frac{0}{x-7}\) simplifies to \(0\) given \(x \neq 7\) as the denominator cannot be zero. The end result is \( 0 \).

Key Concepts

Combining Like Terms

When working with algebraic expressions, combining like terms is a crucial step. Like terms are terms that have identical algebraic variables raised to the same power. In the original exercise, all the fractions share a common denominator \(x-7\), which makes them like terms. Here are some simple pointers to identify and combine like terms effectively:

• Identify terms with the same variable part. In our case, each fraction has the denominator \(x - 7\).
• Add or subtract the coefficients of the like terms as dictated by the expression.
• Place the result over the common denominator if dealing with rational expressions.

In this example, we have \(\frac{2x-1}{x-7}\), \(\frac{3x-1}{x-7}\), and \(\frac{5x-2}{x-7}\) as like terms. By combining these terms, we simplified the fractions without altering the base of the expression.

Rational Expressions

Rational expressions involve fractions where the numerator and the denominator are polynomials. Understanding how to manipulate these expressions is essential in algebra. Key elements to remember when dealing with rational expressions include:

• The denominator is crucial. It cannot be zero since division by zero is undefined.
• Like terms in rational expressions can be combined to simplify the expression.
• Operations such as addition, subtraction, multiplication, and division are similar to regular fractions.

In the given exercise, the presence of \(\frac{2x-1}{x-7}\), \(\frac{3x-1}{x-7}\), and \(\frac{5x-2}{x-7}\) shows us a good example of working with rational expressions. The exercise demonstrates the simplification process using the common denominator so that the answer becomes simple and manageable.

Simplifying Expressions

Simplifying expressions involves reducing a complex expression to its simplest form. This can be achieved by combining like terms, reducing fractions, and canceling out variables where possible. Here’s a simple guide to follow when simplifying expressions:

• Look for and combine like terms.
• Simplify the numerators and denominators separately when dealing with rational expressions.
• Check and cancel terms when possible to simplify further.

For the expression \(\frac{2x-1+3x-1-5x+2}{x-7}\), simplifying the numerator results in \(0\), leaving us with \(\frac{0}{x-7}\). Since any number divided by a non-zero number is 0, our expression simplifies completely to 0. Simplification helps solve the exercise efficiently and aids in understanding the bigger picture of an equation.