Linear Equations in One Variable
Last Updated : 09 Jan, 2025
Linear equation in one variable is the equation that is used for representing the conditions that are dependent on one variable. It is a linear equation i.e. the equation in which the degree of the equation is one, and it only has one variable.
A linear equation in one variable is a mathematical statement that involves a first-degree polynomial, and it can be expressed in the form: ax + b = c
Example: We can take any variable such as x, y, a, b, etc. Some examples of linear equations in one variable are,
- 2a + 3a = 20
- 5x + x = 12, etc.
The above equations are linear in one variable as they only have one variable and the highest degree of the variable is 1.
Graph of Linear Equation in One Variable
These linear equation can be easily represented on the graphs and they represent the straight line which can be horizontal to the coordinate axis or vertical to the coordinate axis.
We represent the mathematical condition using these equations as the condition "a number which is 5 less than twice of itself" is represented by the linear equation,
x = 2x - 5 , Where x is the unknown number.
In the above equation there is only one variable (x) and the degree of the variable is one thus, it is a linear equation in one variable.
A linear equation in one variable can be expressed in the standard form
ax+b = 0
where x is the variable and a and b are the constants involved. These constants (a and b) are non-zero real numbers. These equations have only one possible solution for the value of the variable.
Solving Linear Equations in One Variable
Linear Equations in One Variable can easily be solved by following the steps discussed below,
- Step 1: Write the given equation in standard form and if a or b is a fraction then take the LCM of the fraction to make them integers.
- Step 2: The constants are then taken to the right side of the equation.
- Step 3: All the variables and the constant terms are then simplified to form one single variable term and one single constant.
- Step 4: The coefficient of the variable is made 1 by dividing both sides with a suitable constant to get the final result.
Let's understand the above steps with the help of the following example.
Examples on Linear Equation in One Variable
Solve the following linear equation in one variable.
Example: Solve 3x + 1/2 = (1/2)x - 13/2
Solution:
Step 1: Arranging in standard form and taking LCM
3x - (1/2)x + 1/2 + 13/2 = 0
⇒ (6x - x + 1 + 13)/2 = 0
⇒ 6x - x + 1 + 13 = 0
Step 2: Transposing the constant to the right-hand side
6x - x = -1 - 13
Step 3: Simplification
5x = -14
Step 4: Make coefficient of x to 1
Dividing both sides by 5 we get,
5x/5 = -14/5
x = -14/5
This is the required solution.
Linear Equations and Non-Linear Equations
As we have already learned that linear equations are the equations with the highest degree of "one" and they represent a straight line on the coordinate plane. Examples of linear equations are,
- 3x = 5
- x + y = 12
- 3x - 4y = 11, etc.
All these linear equations represent the straight line in coordinate planes.
All the equations which have a degree greater than one are called non-linear equations they represent curves in the coordinate plane. Various types of non-linear equations represent various types of curves in the 2-D plane. Some of the common curves which we study are,
Circle: x2 + y2 = 49, this equation represents a circle in an x-y coordinate plane with a center at (0, 0) and a radius of 7 units.
Similarly,
All these equations discussed above are non-linear equations.
Difference between Linear Equations and Non-Linear Equations
The key differences between Linear Equations and Non-Linear Equations are:
| Feature | Linear Equations | Non-Linear Equations |
|---|
| Definition | Equations of the form ax + by = c, where a, b, and c are constants, and x and y are variables raised to the power of 1. | Equations where the highest power of the variables is greater than 1, or variables are multiplied together. |
| Graph | Represented by straight lines. | Represented by curves or irregular shapes. |
| Solution(s) | Always forms a straight line when graphed. | Can form curves, loops, or irregular shapes when graphed. |
| Number of Solutions | Always one solution (if the lines intersect). | Can have multiple solutions or none. |
| Solution Method | Solvable using algebraic methods like substitution, elimination, or graphing. | Often require numerical or iterative methods for solutions. |
Must Read
Also Check
Linear Equation in One Variable Examples
Example 1: Solve for y, 8y - 4 = 0
Solution:
Solving for value of y,
Adding 4 to both sides of the equation ,
8y -4 + 4 = 4
8y = 4
Dividing both sides of equation by 8
y = 4/8
Simplifying the equation ,
y = 1/2
Example 2: Solve the equation in x, 3x +10 = 55
Solution:
Taking constants to RHS,
3x = 45
x = 15
Example 3: Solve the equation in x, 4/x5 -5 = 15
Solution:
4/x5 -5 = 15
⇒ 4x/5 = 15 + 5
⇒ 4x/5 = 20
⇒ x = 20×5/4
⇒ x = 25
Example 4: The age of Ravi is twice the age of his sister Kiran if the sum of their age is 24 find their individual age.
Solution:
Let the age of Kiran is x, then the age of Ravi is 2x
Given, the sum of their ages is 24
x + 2x = 24
⇒ 3x = 24
⇒ x = 8
Thus, the age of Kiran = x = 8 years
Age of Ravi = 2x = 2×8 = 16 years
Example 5: Akshay earns three times more than Abhay and if the difference in their salaries is Rs 5000 find their individual salaries.
Solution:
Let the salary of Abhay be x, then the salary of Akshay is 3x
Given, the difference in their salaries is Rs 5000
3x - x = 5000
⇒ 2x = 5000
⇒ x = 2500
Thus, the salary of Abhay = x = Rs 2500
The salary of Akshay = 3x = 3×2500 = Rs 7500
Similar Reads
CBSE Class 8th Maths Notes CBSE Class 8th Maths Notes cover all chapters from the updated NCERT textbooks, including topics such as Rational Numbers, Algebraic Expressions, Practical Geometry, and more. Class 8 is an essential time for students as subjects become harder to cope with. At GeeksforGeeks, we provide easy-to-under
15+ min read
Chapter 1: Rational Numbers
Rational NumbersA rational number is a type of real number expressed as p/q, where q ≠0. Any fraction with a non-zero denominator qualifies as a rational number. Examples include 1/2, 1/5, 3/4, and so forth. Additionally, the number 0 is considered a rational number as it can be represented in various forms such a
9 min read
Natural Numbers | Definition, Examples & PropertiesNatural numbers are the numbers that start from 1 and end at infinity. In other words, natural numbers are counting numbers and they do not include 0 or any negative or fractional numbers.Here, we will discuss the definition of natural numbers, the types and properties of natural numbers, as well as
11 min read
Whole Numbers - Definition, Properties and ExamplesWhole numbers are the set of natural numbers (1, 2, 3, 4, 5, ...) plus zero. They do not include negative numbers, fractions, or decimals. Whole numbers range from zero to infinity. Natural numbers are a subset of whole numbers, and whole numbers are a subset of real numbers. Therefore, all natural
10 min read
Integers | Definition, Examples & TypesThe word integer originated from the Latin word “Integer†which means whole or intact. Integers are a special set of numbers comprising zero, positive numbers, and negative numbers. So, an integer is a whole number (not a fractional number) that can be positive, negative, or zero. Examples of intege
8 min read
Rational NumbersRational numbers are a fundamental concept in mathematics, defined as numbers that can be expressed as the ratio of two integers, where the denominator is not zero. Represented in the form p/q​ (with p and q being integers), rational numbers include fractions, whole numbers, and terminating or repea
15+ min read
Representation of Rational Numbers on the Number Line | Class 8 MathsRational numbers are the integers p and q expressed in the form of p/q where q>0. Rational numbers can be positive, negative or even zero. Rational numbers can be depicted on the number line. The centre of the number line is called Origin (O). Positive rational numbers are illustrated on the righ
5 min read
Rational Numbers Between Two Rational Numbers | Class 8 MathsReal numbers are categorized into rational and irrational numbers respectively. Given two integers p and q, a rational number is of the form p/q, where q > 0. A special case arises when q=1 and the rational number simply becomes an integer. Hence, all integers are rational numbers, equal to p. Th
6 min read
Chapter 2: Linear Equations in One Variable
Algebraic Expressions in Math: Definition, Example and EquationAlgebraic Expression is a mathematical expression that is made of numbers, and variables connected with any arithmetical operation between them. Algebraic forms are used to define unknown conditions in real life or situations that include unknown variables.An algebraic expression is made up of terms
8 min read
Linear Equations in One VariableLinear equation in one variable is the equation that is used for representing the conditions that are dependent on one variable. It is a linear equation i.e. the equation in which the degree of the equation is one, and it only has one variable.A linear equation in one variable is a mathematical stat
7 min read
Linear Equations in One Variable - Solving Equations which have Linear Expressions on one Side and Numbers on the other Side | Class 8 MathsLinear equation is an algebraic equation that is a representation of the straight line. Linear equations are composed of variables and constants. These equations are of first-order, that is, the highest power of any of the involved variables i.e. 1. It can also be considered as a polynomial of degre
4 min read
Solving Linear Equations with Variable on both SidesEquations consist of two main components: variables and numbers. Understanding the relationship between these components and how to manipulate them is essential for solving equations.Variable: A variable is a symbol (often a letter like x, y, or z) that represents an unknown or changing quantity.Num
6 min read
Reducing Equations to Simpler Form | Class 8 MathsReducing equations is a method used to simplify complex equations into a more manageable form. This technique is particularly useful when dealing with non-linear equations, which cannot always be solved directly. By applying specific mathematical operations, such as cross-multiplication, these equat
6 min read
Equations Reducible to Linear FormEquations Reducible to Linear Form" refers to equations that can be transformed or rewritten into a linear equation. These equations typically involve variables raised to powers other than 1, such as squared terms, cubed terms, or higher. By applying suitable substitutions or transformations, these
9 min read
Chapter 3: Understanding Quadrilaterals
Types of PolygonsTypes of Polygons classify all polygons based on various parameters. As we know, a polygon is a closed figure consisting only of straight lines on its edges. In other words, polygons are closed figures made up of more than 2 line segments on a 2-dimensional plane. The word Polygon is made up of two
9 min read
Triangles in GeometryA triangle is a polygon with three sides (edges), three vertices (corners), and three angles. It is the simplest polygon in geometry, and the sum of its interior angles is always 180°. A triangle is formed by three line segments (edges) that intersect at three vertices, creating a two-dimensional re
13 min read
QuadrilateralsQuadrilateral is a two-dimensional figure characterized by having four sides, four vertices, and four angles. It can be broadly classified into two categories: concave and convex. Within the convex category, there are several specific types of quadrilaterals, including trapezoids, parallelograms, re
12 min read
Area of PentagonArea of Pentagon or the area of any polygon is the total space taken by that geometric object. In geometry area and perimeter are the most fundamental quantities of measurement after the side. In general, we study two types of shapes in geometry one is flat shapes(2-D Shapes) and other solid shapes
7 min read
Sum of Angles in a PolygonPolygon is defined as a two-dimensional geometric figure that has a finite number of line segments connected to form a closed shape. The line segments of a polygon are called edges or sides, and the point of intersection of two edges is called a vertex. The angle of a polygon is referred to as the s
11 min read
Exterior Angles of a PolygonPolygon is a closed, connected shape made of straight lines. It may be a flat or a plane figure spanned across two-dimensions. A polygon is an enclosed figure that can have more than 3 sides. The lines forming the polygon are known as the edges or sides and the points where they meet are known as ve
6 min read
Trapezium: Types | Formulas |Properties & ExamplesA Trapezium or Trapezoid is a quadrilateral (shape with 4 sides) with exactly one pair of opposite sides parallel to each other. The term "trapezium" comes from the Greek word "trapeze," meaning "table." It is a two-dimensional shape with four sides and four vertices.In the figure below, a and b are
8 min read
Kite - QuadrilateralsA Kite is a special type of quadrilateral that is easily recognizable by its unique shape, resembling the traditional toy flown on a string. In geometry, a kite has two pairs of adjacent sides that are of equal length. This distinctive feature sets it apart from other quadrilaterals like squares, re
8 min read
Parallelogram | Properties, Formulas, Types, and TheoremA parallelogram is a two-dimensional geometrical shape whose opposite sides are equal in length and are parallel. The opposite angles of a parallelogram are equal in measure and the Sum of adjacent angles of a parallelogram is equal to 180 degrees.A parallelogram is a four-sided polygon (quadrilater
10 min read
Properties of ParallelogramsProperties of Parallelograms: Parallelogram is a quadrilateral in which opposite sides are parallel and congruent and the opposite angles are equal. A parallelogram is formed by the intersection of two pairs of parallel lines. In this article, we will learn about the properties of parallelograms, in
9 min read
Rhombus: Definition, Properties, Formula and ExamplesA rhombus is a type of quadrilateral with the following additional properties. All four sides are of equal length and opposite sides parallel. The opposite angles are equal, and the diagonals bisect each other at right angles. A rhombus is a special case of a parallelogram, and if all its angles are
6 min read
Square in Maths - Area, Perimeter, Examples & ApplicationsA square is a type of quadrilateral where all four sides are of equal length and each interior angle measures 90°. It has two pairs of parallel sides, with opposite sides being parallel. The diagonals of a square are equal in length and bisect each other at right angles.Squares are used in various f
5 min read
Chapter 4: Practical Geometry
Chapter 5: Data Handling
Data HandlingData Handling: Nowadays, managing and representing data systematically has become very important especially when the data provided is large and complex, This is when Data Handling comes into the picture.Data handling involves the proper management of research data throughout and beyond the lifespan
12 min read
What is Data Organization?It is a critical process that involves structuring, categorizing, and managing data to make it more accessible, usable, and analyzable. Whether in research, business, or everyday applications, well-organized data can significantly enhance efficiency and decision-making. The importance of data organi
9 min read
Frequency Distribution - Table, Graphs, FormulaA frequency distribution is a way to organize data and see how often each value appears. It shows how many times each value or range of values occurs in a dataset. This helps us understand patterns, like which values are common and which are rare. Frequency distributions are often shown in tables or
11 min read
Pie ChartPie chart is a popular and visually intuitive tool used in data representation, making complex information easier to understand at a glance. This circular graph divides data into slices, each representing a proportion of the whole, allowing for a clear comparison of different categories making it ea
11 min read
Chance and ProbabilityChance is defined as the natural occurrence of any event without any interference, we can also say that the possibility of any event is the chance of the event, and mathematically we define the chance as the probability of an event.Probability refers to the likelihood of the occurrence of an event.
9 min read
Random Experiment - ProbabilityIn a cricket match, before the game begins. Two captains go for a toss. Tossing is an activity of flipping a coin and checking the result as either “Head†or “Tailâ€. Similarly, tossing a die gives us a number from 1 to 6. All these activities are examples of experiments. An activity that gives us a
11 min read
Probability in MathsProbability is the branch of mathematics where we determine how likely an event is to occur. It is represented as a numeric value ranging from 0 to 1. Probability can be calculated as:\text{Probability} = \dfrac{Favourable \ Outcome}{Total \ Number \ of \ Outcomes}Favourable outcomes refer to the ou
4 min read
Chapter 6: Squares and Square Roots
Chapter 7: Cubes and Cube Roots
Chapter 8: Comparing Quantities
Ratios and PercentagesRatios and Percentages: Comparing quantities is easy, each of the quantities is defined to a specific standard and then the comparison between them takes place after that. Comparing quantities can be effectively done by bringing them to a certain standard and then comparing them related to that spec
6 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
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
Discount FormulaDiscount in Mathematics is defined as the reduction in price of any service and product. Discount is offered by the business owner to easily and quickly sell their product or services. Giving discounts increases the sales of the business and helps the business retain its customer. Discount is always
9 min read
Sales Tax, Value Added Tax, and Goods and Services Tax - Comparing Quantities | Class 8 MathsTax is a mandatory fee levied by the government to collect revenue for public works providing the best facilities and infrastructure.The first known Tax system was in Ancient Egypt around 3000–2800 BC, in First Dynasty of Egypt. The first form of taxation was corvée and tithe. In India, The Tax was
5 min read
Simple InterestSimple Interest (SI) is a method of calculating the interest charged or earned on a principal amount over a fixed period. It is calculated based solely on the principal amount, which remains unchanged throughout the calculation.Simple Interest is widely used across industries such as banking, financ
9 min read
Compound Interest | Class 8 MathsCompound Interest: Compounding is a process of re-investing the earnings in your principal to get an exponential return as the next growth is on a bigger principal, following this process of adding earnings to the principal. In this passage of time, the principal will grow exponentially and produce
9 min read
Compound InterestCompound Interest is the interest that is calculated against a loan or deposit amount in which interest is calculated for the principal as well as the previous interest earned. Compound interest is used in the banking and finance sectors and is also useful in other sectors. A few of its uses are:Gro
9 min read
Chapter 9: Algebraic Expressions and Identities
Algebraic Expressions and IdentitiesAn algebraic expression is a mathematical phrase that can contain numbers, variables, and operations, representing a value without an equality sign. Whereas, algebraic identities are equations that hold true for all values of the variables involved. Learning different algebraic identities is crucial
10 min read
Types of Polynomials (Based on Terms and Degrees)Types of Polynomials: In mathematics, an algebraic expression is an expression built up from integer constants, variables, and algebraic operations. There are mainly four types of polynomials based on degree-constant polynomial (zero degree), linear polynomial ( 1st degree), quadratic polynomial (2n
9 min read
Like and Unlike Algebraic Terms: Definition and ExamplesLike terms are terms in algebraic expressions that have the same variables raised to the same powers. Like and Unlike Terms are the types of terms in algebra, and we can differentiate between like and unlike terms by simply checking the variables and their powers. We define algebraic terms as the in
7 min read
Mathematical Operations on Algebraic Expressions - Algebraic Expressions and Identities | Class 8 MathsThe basic operations that are being used in mathematics (especially in real number systems) are addition, subtraction, multiplication and so on. These operations can also be done on the algebraic expressions. Let us see them in detail. Algebraic expressions (also known as algebraic equations) are de
5 min read
Multiplying PolynomialsPolynomial multiplication is the process of multiplying two or more polynomials to find their product. It involves multiplying coefficients and applying exponent rules for variables.When multiplying polynomials:Multiply the coefficients (numerical values).Multiply variables with the same base by add
8 min read
Standard Algebraic IdentitiesAlgebraic Identities are algebraic equations that are always true for every value of the variable in them. The algebraic equations that are valid for all values of variables in them are called algebraic identities. It is used for the factorization of polynomials. In this way, algebraic identities ar
7 min read