Problem 44 - Algebra Form and Function

Question

Is the given expression linear in the indicated variable? Assume all constants are non-zero. $$ 3 x y+b x+2-10 y, y $$

Step-by-Step Solution

Answer

Answer: Yes, the expression is linear with respect to the variable y.

Step 1: Identify the given expression and variable

We are given the expression: $$ 3xy + bx + 2 - 10y $$ with respect to the variable y.

Step 2: Analyze the expression for linearity with respect to y

To determine if the expression is linear in y, we need to rewrite the expression as a product and a sum/difference of terms involving y and terms not involving y: $$ (3x - 10)y + bx + 2 $$ Now, let's inspect the expression.

Step 3: Inspect the terms with the variable y

We have only one term with the variable y: $$ (3x - 10)y $$ This term is in the form of "constant × y" where the constant is \((3x - 10)\).

Step 4: Inspect the terms without the variable y

We have two terms that do not involve y: $$ bx + 2 $$ These terms do not contain the variable y and are constants with respect to y.

Step 5: Determine if the expression is linear in y

Given that the expression is in the form of "constant × y + any other term without y" as \((3x - 10)y + bx + 2\), we can conclude that the expression is linear with respect to the variable y.

Key Concepts

Variable Analysis

When working with algebraic expressions, understanding the role of variables is crucial. A variable is a symbol, like \( x \) or \( y \), used to represent a number in mathematical expressions or equations. In our example expression, \( 3xy + bx + 2 - 10y \), we have been asked to focus on the variable \( y \).

To analyze variables, we identify which parts of the expression involve the variable in question. For our expression:

• The term \( 3xy \) includes \( y \) alongside \( x \).
• The term \( -10y \) directly involves \( y \).
• The term \( bx + 2 \) does not involve \( y \).

This approach helps us when determining other behaviors such as linearity.

Constant Identification

In any algebraic equation, constants are the fixed values that don't change when the variable's value changes. Identifying constants is key to understanding the structure of expressions.

In our example, the expression is \( 3xy + bx + 2 - 10y \). Let's break this down:

• The constant in the term \( b x + 2 \) is \( 2 \), itself a standalone constant.
• The term \( 3xy \) involves the constant \( 3 \) when considering its effect on \( y \).
• The expression \( (3x - 10)y \) introduces the role of \( 10 \) as a constant part of the coefficient affecting \( y \).

Being able to spot these constants helps us simplify and analyze expressions efficiently.

Linearity Determination

A key question in algebra is whether an expression is linear with respect to a specific variable. For clarity, an expression is linear in a variable if it can be expressed in the form: \[ a \, imes \, \text{variable} + \text{constant term}\] This setup also means the variable is raised only to the first power.

In our expression \( 3xy + bx + 2 - 10y \), we consider linearity concerning \( y \). We rewrite it in a rearranged form:\[(3x - 10)y + bx + 2\]Here, \((3x - 10)\) acts as a collective "constant" multiplying \( y \), satisfying the principle of linearity. The remaining terms, \( bx + 2 \), don't include \( y \), aligning with linear expression criteria.

• Only terms with "constant × variable" are considered.
• All terms without the variable are treated as constants for that variable.

Therefore, according to the definition, the expression is indeed linear with respect to \( y \).