Problem 101 - Algebra and Trigonometry

Question

determine whether each statement makes sense or does not make sense, and explain your reasoning. I subtracted \(\frac{3 x-5}{x-1}\) from \(\frac{x-3}{x-1}\) and obtained a constant.

Step-by-Step Solution

Answer

The statement makes sense because the result of this subtraction is indeed a constant, 2, when \(x ≠ 1\).

Step 1: Simplify the given expressions

Given the expressions \(\frac{3 x-5}{x-1}\) and \(\frac{x-3}{x-1}\), first simplify both by subtracting the numerators since they have the same denominators. This gives \(\frac{3 x-5 - (x-3)}{x-1}\).

Step 2: Perform the subtraction in the numerator

The numerator simplifies as follows: \(3x - 5 - x + 3\), which gives \(2x - 2\) . Therefore, the new expression now is \(\frac{2x - 2}{x - 1}\).

Step 3: Factorise and Simplify

Factor 2 out of the new expression to get \(\frac{2(x - 1)}{x - 1}\). You'll notice that the \(x - 1\) terms in the numerator and denominator will cancel out as long as \(x ≠ 1\), giving a constant, 2.

Key Concepts

Simplifying Expressions

Simplifying expressions involves making an algebraic expression easier to work with. When dealing with rational expressions, like fractions, simplification is often about ensuring numerators and denominators have no like terms that can be combined or further simplified. In this exercise, the simplification begins by focusing on expressions with a common denominator. Both fractions, \(\frac{3x-5}{x-1}\) and \(\frac{x-3}{x-1}\), share the denominator \(x-1\). This means subtraction can happen directly with the numerators. This move is key to simplifying as it prepares the expression for further operations. Remember, having a common denominator is the gateway in handling operations like subtraction in rational expressions.

Factoring

Factoring is breaking down an expression into components that when multiplied together will give the original expression back. It's a vital step in many algebraic operations, making expressions simpler or preparing them for simplification or evaluation. In the example of subtracting rational expressions, after combining the numerators, we got \(2x - 2\). Factoring this is crucial; you find the greatest common factor (GCF), which here is 2. So, \(2x - 2\) can be rewritten as \(2(x - 1)\). This transformation is important for further simplification and for identifying any factors common to both the numerator and denominator. Factoring not only simplifies expressions but sometimes turns them into forms we can easily understand or manipulate further.

Canceling Common Factors

Canceling common factors is an essential step after simplifying and factoring expressions. It helps reduce the expression to its simplest form. Once the expression \(\frac{2(x - 1)}{x - 1}\) is formed by factoring, you can notice something immediate and powerful: the \((x - 1)\) factors appear in both the numerator and the denominator. These similar terms can be canceled or divided out, as long as \(x eq 1\), which would make the denominator zero. This process results in reducing the expression to a constant. Canceling is a technique often used in algebra to simplify expressions further, making calculations more straightforward and results easier to interpret.

Subtraction in Algebra

Subtraction is one of the basic operations in algebra, and handling subtraction of rational expressions follows specific rules. It involves aligning terms based on their denominators when dealing with fractions. In our provided exercise, subtraction involves rational expressions with the same denominator, which simplifies the process. The numerators subtract directly: \(3x - 5 - (x-3)\). This systematic process ensures the subtraction is accurate and adheres to algebraic principles.Understanding subtraction in algebra helps see how expressions combine or reduce, and prepares the way for simplifying and factoring, as highlighted in previous sections. Recognizing these steps enhances comprehension and empowers anyone tackling similar algebraic expressions to do so more confidently and effectively.