A coefficient is a number that multiplies a variable in a mathematical expression. It tells you how much of that variable you have. For example, in the term 5x, the coefficient is 5 — it means 5 times the variable x.
Coefficients can be positive, negative, or zero.
Algebraic EquationA coefficient is a scalar value that indicates the variable's impact on an expression. When a variable in an expression has no written coefficient, it is assumed to be one, because multiplying by 1 does not change its value.
For example, in the given expression: 10x + x2 + 7, it has two coefficients:
- 10, which is the coefficient of x.
- 1, which is the coefficient of x2, as it doesn't have a number with it, we automatically assume it to be one.
- 7 is the constant.
They help in understanding the relationship between different parts of the expression. In simple terms, coefficients tell us how much one quantity affects another within a mathematical equation or formula.
Properties of Coefficients
Properties of coefficients in mathematics include:
- Linearity: Coefficients exhibit linearity, meaning they distribute over addition and subtraction. For example, in (ax + by), the coefficient (a) multiplies (x), and (b) multiplies (y).
- Commutativity: The order of coefficients and variables does not affect the result when multiplying. For example, 2 × y and y × 2 both represent the product of 2 and y.
- Associativity: Coefficients are associative with multiplication. For instance, in (2 × 3x), the result is the same as (3 × 2x), yielding (6x) in both cases.
- Identity Property: Coefficient (1) serves as the identity element in multiplication. Multiplying any variable by (1) leaves the variable unchanged.
- Additive Identity: Adding (0) as a coefficient does not alter the value of the expression. For example, (3x + 0 = 3x).
- Zero Coefficient: A coefficient of (0) nullifies the variable's contribution to the expression. For instance, (0x = 0)
Coefficient of a Variable
The coefficient of a variable is a number that is multiplied by the variable in an algebraic expression. For example, in the expression 5x, the coefficient of the variable x is 5. Here, 5 is the coefficient, and x is the variable.
Similarly, in the expression 3y, the coefficient of the variable y is -3. The coefficient indicates how many times the variable is multiplied by itself or by another term in the expression.
Numerical Coefficient
The coefficient is a number that is multiplied by a variable in an algebraic expression. For example, in the expression 3xy, the numerical coefficient is 3. Here, "3" is multiplied by the variable "xy." Similarly, in the expression -2y, the numerical coefficient is -2. The numerical coefficient indicates the scale or magnitude of the variable's effect on the expression.
Leading Coefficient
The Leading coefficient is the coefficient of the term with the highest degree in a polynomial expression. For example, in the polynomial (3x3 - 5x2 + 2x + 1), the leading coefficient is 4 because it is attached to the term (x3), which has the highest degree (3) among all the terms.
Coefficient vs Constant
The difference between a coefficient and a constant can be understood by the table given below:
Coefficient | Constant |
|---|
A numerical factor multiplying a variable in an expression | A fixed numerical value in an expression |
Indicates the scale or magnitude of the variable's effect | Remains constant, independent of variables |
Multiplied by variables in the expression | Not multiplied by variables, stands alone |
It can vary based on the term and variables involved | Does not change within the context of expression |
In 3x+2y, coefficients are 3 and 2 for x and y | In 4x+7, the constant is 7 |
How to Find a Coefficient?
To find a coefficient in an algebraic expression, follow these steps:
Step 1: Identify the term containing the variable for which you want to find the coefficient.
Step 2: Examine the numerical value directly attached to the variable within that term.
That numerical value is the coefficient of the variable in the expression.
For example, in the expression 4x + 3y - 2
- If you want to find the coefficient of x, locate the term 4x. Then the coefficient of x is 4.
- If you want to find the coefficient of y, locate the term 3y. Then the coefficient of y is 3.
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Solved Examples on Coefficient of a Variable
Some of the examples of the Coefficient of Variable are discussed below:
Example 1: In the expression 5x-2y+3z, what are the coefficients of x, y, and z?
Solution:
In the expression 5x - 2y + 3z, the coefficients are as follows:
- Coefficient of x: Coefficient of x is the number directly multiplied by x, which is 5.
- Coefficient of y: Coefficient of y is the number directly multiplied by y, which is -2. (Note: Coefficients can be negative.)
- Coefficient of z: Coefficient of z is the number directly multiplied by z, which is 3.
So, coefficients of x, y, and z are 5, -2, and 3 respectively.
Example 2: A company produces two types of products, A and B. The profit from selling each unit of product A is $3, and the profit from selling each unit of product B is $5. If the company sells x units of product A and y units of product B, write an expression to represent the total profit.
Solution:
To represent the total profit, we need to multiply the number of units sold for each product by their respective profits and then sum the results.
Here's the expression:
Total profit = (3x + 5y)
Expression represents the profit from selling (x) units of product A, each yielding $3 profit, and (y) units of product B, each yielding $5 profit.
Suppose the company sells 10 units of product A (x = 10) and 15 units of product B (y = 15).
Putting these values into the expression:
Total profit = (3 × 10 + 5 × 15)
= (30 + 75)
= 105
So, if the company sells 10 units of product A and 15 units of product B, the total profit would be $105.
Example 3: Solve the equation 2x + 4 = 10 to find the value of x.
Solution:
To solve the equation 2x + 4 = 10 for x, follow these steps:
Isolate the variable term: Subtract 4 from both sides of the equation to isolate the term containing x:
2x + 4 − 4 = 10−4
2x = 6
Solve for x: Divide both sides by 2 to solve for x:
2x/3 = 6/2
x = 3
So, the value of x that satisfies the equation 2x + 4 = 10 is x = 3.
Practice Questions on Coefficient
Some Practice Questions on Coefficient are,
Question 1: The Perimeter of a rectangle is 10x + 6, where x represents the length of one side of the rectangle. If the width of the rectangle is 2x, find the expression for the length.
Question 2: Factor the expression 4x2 + 12x completely.
Question 3: Temperature T in degrees Celsius is given by the formula T = 5x + 32, where x is the temperature in degrees Fahrenheit. If the temperature outside is 20°F, what is the corresponding temperature in degrees Celsius?
Question 4: Evaluate the expression 2x3 - 3x2 + x - 4 for x = 2.
Question 5: A charity organization collects donations from two sources: individuals and corporations. For every dollar donated by an individual, the charity receives $0.75, and for every dollar donated by a corporation, the charity receives $0.90. If x represents the amount donated by individuals and y represents the amount donated by corporations, write an expression to represent the total amount received by the charity.