Problem 15 - Elementary Algebra

Question

In the term \(3 x(y-1)\), the coefficient of (a) \(x(y-1)\) is __________. (b) \((y-1)\) is _________. (c) \(3(y-1)\) is ______. (d) \(x\) is _______. (e) 3 is ________. (f) The numerical coefficient is _________.

Step-by-Step Solution

Answer

Answer: (a) Coefficient of \(x(y-1)\): 3 (b) Coefficient of \((y-1)\): \(3x\) (c) Coefficient of \(3(y-1)\): x (d) Coefficient of x: \(3(y-1)\) (e) Coefficient of 3: 1 (f) Numerical coefficient: 3

Step 1: (a) Coefficient of \(x(y-1)\)

The term \(x(y-1)\) is being multiplied by the scalar 3, so the coefficient of \(x(y-1)\) is 3.

Step 2: (b) Coefficient of \((y-1)\)

Within the term \(3x(y-1)\), the term \((y-1)\) is being directly multiplied by both 3 and x. Therefore, the coefficient of \((y-1)\) is \(3x\).

Step 3: (c) Coefficient of \(3(y-1)\)

The expression \(3(y-1)\) is a part of the original expression \(3x(y-1)\). The term \(3(y-1)\) is being multiplied by x, so the coefficient of \(3(y-1)\) is x.

Step 4: (d) Coefficient of x

In the expression \(3x(y-1)\), x is being multiplied by 3 and \((y-1)\). Therefore, the coefficient of x is \(3(y-1)\).

Step 5: (e) Coefficient of 3

The number 3 is multiplying the term \(x(y-1)\) in the expression \(3x(y-1)\). As 3 is just a constant scalar, the coefficient of 3 is said to be 1 as multiplying any scalar with 1 doesn't change its value (identity property of multiplication).

Step 6: (f) Numerical coefficient

The numerical coefficient refers to the constant scalar that is multiplying the variables in an expression. In this case, the numerical coefficient is 3, as it's the constant scalar that multiplies the term \(x(y-1)\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. For example, in the expression 3x(y-1), there are several components – a numerical coefficient (3), variables (x and y), and an arithmetic operation (subtraction in the parentheses). Understanding the structure of algebraic expressions is crucial for solving equations and simplifying complex mathematical terms.

When we deal with algebraic expressions, it's important to identify the different parts. The expression 3x(y-1) can be seen as a product of 3 and x(y-1). This product format often leads us to discuss terms like coefficients and constants, which play critical roles in algebra.

For the exercises similar to the example given, it's important to notice how each component interacts with the others. The numerical coefficient (3) is right next to the variable x, implying multiplication. Inside the parentheses, the subtraction operation affects the term as a whole, but does not change the coefficient for any individual part of the expression.

Numerical Coefficients

Numerical coefficients are constants that multiply a variable or a set of variables within an algebraic expression. They are usually numbers and provide a scaling factor for the variables they are attached to. For instance, in the term 3x, the number '3' is the numerical coefficient, showing that the variable x is to be multiplied by '3'.

In the context of the given problem, the numerical coefficient is particularly easy to spot; it is the number '3' that precedes the term x(y-1). This '3' scales the whole expression, affecting how it interacts with other terms if it were part of a larger equation or expression. When students identify these coefficients, they should interpret them as the 'weight' given to the variable parts of an expression. It's also worth noting that if there’s no numerical value written before a variable, it's implied that the coefficient is 1.

Variable Coefficients

Variable coefficients, unlike numerical coefficients, include variables and sometimes, parentheses to indicate more complex relationships. In our example, considering the term (y-1), the coefficient isn't a simple number but rather the product of '3' and 'x', represented as 3x. This shows that coefficients themselves can be algebraic expressions.

The variable coefficient provides information about how different variables in an expression are connected. It's essential to recognize that while numerical coefficients scale a term, variable coefficients introduce a dependency between terms. In the exercise, understanding that x is the coefficient of 3(y-1) implies that the value of 3(y-1) is dependent on the value of x.

Keep in mind that coefficients aren't just numbers thrown in front of variables for appearance - they play a fundamental role in the behavior of the expression when it undergoes operations such as addition, subtraction, multiplication, or division.