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.
Also Check
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.
Similar Reads
Basic Arithmetic
What are Numbers?Numbers are symbols we use to count, measure, and describe things. They are everywhere in our daily lives and help us understand and organize the world.Numbers are like tools that help us:Count how many things there are (e.g., 1 apple, 3 pencils).Measure things (e.g., 5 meters, 10 kilograms).Show or
15+ min read
Arithmetic OperationsArithmetic Operations are the basic mathematical operations—Addition, Subtraction, Multiplication, and Division—used for calculations. These operations form the foundation of mathematics and are essential in daily life, such as sharing items, calculating bills, solving time and work problems, and in
9 min read
Fractions - Definition, Types and ExamplesFractions are numerical expressions used to represent parts of a whole or ratios between quantities. They consist of a numerator (the top number), indicating how many parts are considered, and a denominator (the bottom number), showing the total number of equal parts the whole is divided into. For E
7 min read
What are Decimals?Decimals are numbers that use a decimal point to separate the whole number part from the fractional part. This system helps represent values between whole numbers, making it easier to express and measure smaller quantities. Each digit after the decimal point represents a specific place value, like t
10 min read
ExponentsExponents are a way to show that a number is multiplied by itself many times. They are everywhere in math and science, helping us write big and easily.Think of exponents like a shortcut for repeated multiplication:23 means 2 x 2 x 2 = 8 52 means 5 x 5 = 25So instead of writing the same number many t
8 min read
PercentageIn mathematics, a percentage is a figure or ratio that signifies a fraction out of 100, i.e., A fraction whose denominator is 100 is called a Percent. In all the fractions where the denominator is 100, we can remove the denominator and put the % sign.For example, the fraction 23/100 can be written a
5 min read
Algebra
Variable in MathsA variable is like a placeholder or a box that can hold different values. In math, it's often represented by a letter, like x or y. The value of a variable can change depending on the situation. For example, if you have the equation y = 2x + 3, the value of y depends on the value of x. So, if you ch
5 min read
Polynomials| Degree | Types | Properties and ExamplesPolynomials are mathematical expressions made up of variables (often represented by letters like x, y, etc.), constants (like numbers), and exponents (which are non-negative integers). These expressions are combined using addition, subtraction, and multiplication operations.A polynomial can have one
9 min read
CoefficientA 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
8 min read
Algebraic IdentitiesAlgebraic Identities are fundamental equations in algebra where the left-hand side of the equation is always equal to the right-hand side, regardless of the values of the variables involved. These identities play a crucial role in simplifying algebraic computations and are essential for solving vari
14 min read
Properties of Algebraic OperationsAlgebraic operations are mathematical processes that involve the manipulation of numbers, variables, and symbols to produce new results or expressions. The basic algebraic operations are:Addition ( + ): The process of combining two or more numbers to get a sum. For example, 3 + 5 = 8.Subtraction (−)
3 min read
Geometry
Lines and AnglesLines and Angles are the basic terms used in geometry. They provide a base for understanding all the concepts of geometry. We define a line as a 1-D figure that can be extended to infinity in opposite directions, whereas an angle is defined as the opening created by joining two or more lines. An ang
9 min read
Geometric Shapes in MathsGeometric shapes are mathematical figures that represent the forms of objects in the real world. These shapes have defined boundaries, angles, and surfaces, and are fundamental to understanding geometry. Geometric shapes can be categorized into two main types based on their dimensions:2D Shapes (Two
2 min read
Area and Perimeter of Shapes | Formula and ExamplesArea and Perimeter are the two fundamental properties related to 2-dimensional shapes. Defining the size of the shape and the length of its boundary. By learning about the areas of 2D shapes, we can easily determine the surface areas of 3D bodies and the perimeter helps us to calculate the length of
10 min read
Surface Areas and VolumesSurface Area and Volume are two fundamental properties of a three-dimensional (3D) shape that help us understand and measure the space they occupy and their outer surfaces.Knowing how to determine surface area and volumes can be incredibly practical and handy in cases where you want to calculate the
10 min read
Points, Lines and PlanesPoints, Lines, and Planes are basic terms used in Geometry that have a specific meaning and are used to define the basis of geometry. We define a point as a location in 3-D or 2-D space that is represented using coordinates. We define a line as a geometrical figure that is extended in both direction
14 min read
Coordinate Axes and Coordinate Planes in 3D spaceIn a plane, we know that we need two mutually perpendicular lines to locate the position of a point. These lines are called coordinate axes of the plane and the plane is usually called the Cartesian plane. But in real life, we do not have such a plane. In real life, we need some extra information su
6 min read
Trigonometry & Vector Algebra
Trigonometric RatiosThere are three sides of a triangle Hypotenuse, Adjacent, and Opposite. The ratios between these sides based on the angle between them is called Trigonometric Ratio. The six trigonometric ratios are: sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec).As give
4 min read
Trigonometric Equations | Definition, Examples & How to SolveTrigonometric equations are mathematical expressions that involve trigonometric functions (such as sine, cosine, tangent, etc.) and are set equal to a value. The goal is to find the values of the variable (usually an angle) that satisfy the equation.For example, a simple trigonometric equation might
9 min read
Trigonometric IdentitiesTrigonometric identities play an important role in simplifying expressions and solving equations involving trigonometric functions. These identities, which include relationships between angles and sides of triangles, are widely used in fields like geometry, engineering, and physics. Some important t
10 min read
Trigonometric FunctionsTrigonometric Functions, often simply called trig functions, are mathematical functions that relate the angles of a right triangle to the ratios of the lengths of its sides.Trigonometric functions are the basic functions used in trigonometry and they are used for solving various types of problems in
6 min read
Inverse Trigonometric Functions | Definition, Formula, Types and Examples Inverse trigonometric functions are the inverse functions of basic trigonometric functions. In mathematics, inverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. The inverse trigonometric functions are the inverse functions of basic trigonometric function
11 min read
Inverse Trigonometric IdentitiesInverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. These functions are the inverse functions of basic trigonometric functions, i.e., sine, cosine, tangent, cosecant, secant, and cotangent. It is used to find the angles with any trigonometric ratio. Inv
9 min read
Calculus
Introduction to Differential CalculusDifferential calculus is a branch of calculus that deals with the study of rates of change of functions and the behaviour of these functions in response to infinitesimal changes in their independent variables.Some of the prerequisites for Differential Calculus include:Independent and Dependent Varia
6 min read
Limits in CalculusIn mathematics, a limit is a fundamental concept that describes the behaviour of a function or sequence as its input approaches a particular value. Limits are used in calculus to define derivatives, continuity, and integrals, and they are defined as the approaching value of the function with the inp
12 min read
Continuity of FunctionsContinuity of functions is an important unit of Calculus as it forms the base and it helps us further to prove whether a function is differentiable or not. A continuous function is a function which when drawn on a paper does not have a break. The continuity can also be proved using the concept of li
13 min read
DifferentiationDifferentiation in mathematics refers to the process of finding the derivative of a function, which involves determining the rate of change of a function with respect to its variables.In simple terms, it is a way of finding how things change. Imagine you're driving a car and looking at how your spee
2 min read
Differentiability of a Function | Class 12 MathsContinuity or continuous which means, "a function is continuous at its domain if its graph is a curve without breaks or jumps". A function is continuous at a point in its domain if its graph does not have breaks or jumps in the immediate neighborhood of the point. Continuity at a Point: A function f
11 min read
IntegrationIntegration, in simple terms, is a way to add up small pieces to find the total of something, especially when those pieces are changing or not uniform.Imagine you have a car driving along a road, and its speed changes over time. At some moments, it's going faster; at other moments, it's slower. If y
3 min read
Probability and Statistics
Basic Concepts of ProbabilityProbability is defined as the likelihood of the occurrence of any event. Probability is expressed as a number between 0 and 1, where, 0 is the probability of an impossible event and 1 is the probability of a sure event.Concepts of Probability are used in various real life scenarios : Stock Market :
6 min read
Bayes' TheoremBayes' Theorem is a mathematical formula that helps determine the conditional probability of an event based on prior knowledge and new evidence.It adjusts probabilities when new information comes in and helps make better decisions in uncertain situations.Bayes' Theorem helps us update probabilities
12 min read
Probability Distribution - Function, Formula, TableA probability distribution is a mathematical function or rule that describes how the probabilities of different outcomes are assigned to the possible values of a random variable. It provides a way of modeling the likelihood of each outcome in a random experiment.While a frequency distribution shows
15+ min read
Descriptive StatisticStatistics is the foundation of data science. Descriptive statistics are simple tools that help us understand and summarize data. They show the basic features of a dataset, like the average, highest and lowest values and how spread out the numbers are. It's the first step in making sense of informat
5 min read
What is Inferential Statistics?After learning basic statistics like how data points relate (covariance and correlation) and probability distributions, the next important step is Inferential Statistics. Unlike descriptive statistics, which just summarizes data, inferential statistics helps us make predictions and conclusions about
6 min read
Measures of Central Tendency in StatisticsCentral tendencies in statistics are numerical values that represent the middle or typical value of a dataset. Also known as averages, they provide a summary of the entire data, making it easier to understand the overall pattern or behavior. These values are useful because they capture the essence o
11 min read
Set TheorySet theory is a branch of mathematics that deals with collections of objects, called sets. A set is simply a collection of distinct elements, such as numbers, letters, or even everyday objects, that share a common property or rule.Example of SetsSome examples of sets include:A set of fruits: {apple,
3 min read
Practice