Problem 43 - Algebra Form and Function

Question

Is the given expression linear in the indicated variable? Assume all constants are non-zero. $$ 3 x y+5 x+2-10 y, x $$

Step-by-Step Solution

Answer

Answer: No, the given expression is not linear in the variable x.

Step 1: Analyze the given expression

Firstly, let's write down the expression: $$ 3xy+5x+2-10y $$

Step 2: Identify the terms related to x

Now let's identify the terms that have x in them: - The first term is \(3xy\). This term has both x and y variables together. - The second term is \(5x\). This term has x alone.

Step 3: Check the powers of x and presence of products

In the two terms with x, we'll now check the powers of x and if x appears in a product of variables: - For the term \(3xy\), x is raised to power 1, but it is in a product with y. - For the term \(5x\), x is raised to power 1, and it is not in a product with another variable.

Step 4: Determine if the expression is linear in x

Since one of the terms with x (i.e., \(3xy\)) is a product of variables, this implies that the given expression is not linear in x. So, the given expression is not linear in the variable x.

Key Concepts

Variables

A variable is a symbol used to represent a number in math. You can think of it as a placeholder that can change value, depending on the problem or situation. Variables are often denoted by letters, like \(x\), \(y\), or \(z\). They help us generalize mathematical ideas and focus on patterns rather than numbers alone.

• Why Use Variables? They allow for solving equations where the actual numbers are unknown.
• Flexibility: Variables let us write formulas that can work with any numbers we plug into them. This makes math versatile and widely applicable across different problems.
• Connecting Concepts: Sometimes dealing with a variable, like \(x\), helps us see how different parts of an expression relate to each other.

In the expression \(3xy + 5x + 2 - 10y\), "x" and "y" are variables. They each stand in for real numbers, allowing the expression to represent a variety of scenarios depending on their values.

Products of Variables

A product of variables occurs when two or more variables are multiplied together. This is a key factor in determining whether an expression is linear. If an expression lacks products of variables, it's closer to linearity.

• Multiplying Variables: When you see something like \(xy\), it means \(x\) is multiplied by \(y\).
• Impact on Linearity: A product like \(xy\) makes an expression nonlinear in both \(x\) and \(y\) because their values interact in a more complex way than simple addition or subtraction.

In our example expression \(3xy + 5x + 2 - 10y\), the term \(3xy\) is a product of the variables \(x\) and \(y\). This interaction prevents the expression from being linear in \(x\) or \(y\).

Powers of Variables

The power or exponent of a variable tells us how many times the variable is used in a multiplication. For example, in the expression \(x^2\), \(x\) is used twice in a product: \(x \, \times \, x\). The term's exponent reveals its degree, which affects linearity.

• Linear Terms: If a variable has an exponent of 1, like \(x\) in \(5x\) or \(x\) in \(3xy\), the expression is potentially linear with respect to that variable.
• Nonlinear Impact: When the variable's exponent is anything other than 1, like \(x^2\), it becomes nonlinear.

In analyzing the expression \(3xy + 5x + 2 - 10y\), we note that all the variables have a power of 1. However, due to the product \(3xy\), it does not maintain linearity with respect to \(x\), even though "x" itself is at power 1. The interaction of multiple variables in a single term is what breaks linearity.