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In mathematics, a linear equation is an equation with the highest degree equal to one and can be expressed using either a single variable or two variables. 3x + 12 = 48, 2x + 3y =12, 3x + 12 = 48 are few examples of linear equations. Applications of linear equations are observed to solve a wide range of real-life situations. A few of the real-life applications of linear equations are Geometry problems, Money problems, and distance-rate-time problems using two variables. To handle such real-life problems using algebra, such problems are represented into mathematical statements to illustrate the relationship between unknown variables and known information.
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Key Takeaways: Algebra, Linear Equation, Variables, Constant, Coefficients, Distance-rate-time problems, Geometry
What is Linear Equation?
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A linear equation is an algebraic expression consisting of a variable and equality sign with the highest degree equal to 1. In simple words, a linear equation can be expressed using two different variables and a constant and has a degree of one. ax + by = c is a standard form of expressing a linear equation where ‘x’ and ‘y’ are variables, ‘a’ and ‘b’ are the coefficients and ‘c’ is the constant term.
The following are examples of two linear equations
2x-3y+4=0 (i)
x+7y-1=0 (ii)
with ‘x’ and ‘y’ as variables and ‘4’ and ‘1’ as constants.
Following mentioned are a few examples of non-linear equation:
x2 + 1 (i)
y+y2 (ii)
1+z+z2+z3 (iii)
The above mentioned are a few non-linear equations since the highest power of the individual variable in each of the equations is greater than one which is the primary criteria for any expression to be treated as a linear equation.
Read Also: Geometry Formula
Types of Linear Equations
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Linear Equations can be represented either using a single variable or two variables.
Linear Equation with One Variable
When an equation is represented with one variable and has a degree equal to one is known as a linear equation with one variable. 3x + 5 = 0, 5x +30 = 0 are a few examples of linear equations with one variable where ‘x’ is the single variable used and both the equations have a degree of one.
Also Read: NCERT Solutions for Class 8 Mathematics Chapter 2: Linear Equations in One Variable
Linear Equation with Two Variables
When an equation is represented using two variables and has a degree equal to one is known as a linear equation with two variables. 3x + 5y = 0, 3x + 7y = 42 are a few examples of linear equations with two variables where ‘x’ and ‘y’ are the two variables used and both the equations have a degree of one.
Also read: Pair of linear equation in two variables
Representation of Real Life problems using Linear Equation
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Real Life problems when represented using algebraic equations clearly illustrate the relationship between unknown variables and the information provided. The following steps help to represent a real-life problem using mathematical statements:
- Convert the problem statement into an algebraic expression to represent the problem in an efficient manner.
- Identify the unknowns in the problem statement and represent those using variables.
- Read the problem statement repeatedly to derive useful information and organize them in the form of data, phrases, and keywords.
- Convert the algebraic expression into an equation using the data provided in the problem statement and apply systemic techniques to solve it.
- Retrace the solution obtained from the problem statement and analyze it thoroughly to determine if it satisfies the problem criterion.
Read Also: Three-dimensional geometry
Applications of Linear Equation in Real-Life:
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There are numerous applications of linear equations in real-life. Such real-life situations are handled using algebra by converting them in mathematical statements which clearly help to illustrate the relationship between unknown variables and known information. Below listed are a few examples where the linear equation is put to use to solve real-life situations:
- Finding solutions to age-related problems
- Determination of speed, distance, and time of a moving object.
- Finding solutions to geometry-related problems
- Problems based on money and percentage
- Finding solutions to force and pressure-related problems
- Problems based on work, time, and wages
Read Also: Perimeter of parallelogram
Things to Remember
- A linear equation may consist of one or two variables.
- A linear equation would always have a degree of one.
- The standard form of representing linear equations is ax + by = c, where x and y are variables, x and y are variables, and c is the constant.
- Addition, subtraction, multiplication, and division are the four main arithmetic operations used to solve linear equations.
- Real-life situations are converted into mathematical statements using algebra.
- The most common application of linear equations is solving word or story problems.
- A linear equation can be used to determine the speed, distance, and time of a moving object, calculate the age of a person, work, time and wage can also be determined with the help of a linear equation.
- The slope of a line can also be represented using a linear equation.
Sample Questions
Ques: Rohit is twice as old as his sister Vani. 10 years ago he was thrice as old as Vani. Find their present ages. [5 marks]
Solution: According to the problem, the age of Rohit and Vani is to be determined. Thus their ages are unknown quantities and to be represented using variables. Since Rohit is twice as old as Vani, let us consider, the age of Vani as ‘x’ and that of Rohit as ‘2x’.
10 years ago, the age of Vani would have been ‘x-10’ and Rohit as thrice as old as Vani, thus Rohit would have been 3(x – 10).
According to the problem statement,
2x – 10 = 3(x – 10)
- 2x -10 = 3x – 30
- x = 20
This implies that the present age of Vani is 20 years and that of Rohit is 40 years.
Ques: The sum of two numbers is 50. If one number is 10 more than the other number, determine the two numbers by framing a linear equation. [5 marks]
Solution: According to the problem, the value of two numbers is to be determined. Thus the two numbers are unknown quantities and to be represented using variables. Since one number is 10 more than the other number, let us consider one number to be ‘x’ and the other number as ‘x+10’.
According to the problem statement,
x + x+ 10 = 50
- 2x = 40
- x = 20
This implies value of one number is 20 while value of the other is 30.
Ques: The length of a rectangle is twice its breadth. If its perimeter is 72 meter, find the length and breadth of the rectangle. [4 marks]
Solution: According to the problem, the length and breadth of a rectangle is to be determined. Thus length and breadth are unknown quantities and to be represented using variables. Since the length is twice the breadth, let us consider the length to be ‘x’ and the breadth as ‘2x’.
According to the problem statement,
2(x + 2x) = 72
- 6x = 72
- x = 12
This implies that the breadth of the rectangle is 12 meter and breath is 24 meter.
Ques: Twice one number minus thrice the second equals 2. The sum of the two numbers is equal to 11. Find the numbers. [5 marks]
Solution: According to the problem, the value of two numbers is to be determined. Thus the two numbers are unknown quantities and to be represented using variables. Since twice one number minus thrice the second is equal to 2, let us consider one number to be ‘x’ and the other number as ‘y’.
According to the problem statement,
2x – 3y = 2 ------------------ (i)
x + y = 11 ------------------ (ii)
Multiplying both sides of (ii) with 3, we get
3x + 3y = 33 ------------------ (iii)
On adding (i) and (iii), we get
5x = 35
x = 7
On substituting this value of x in (ii), we get
7 + y = 11
y = 4
This implies value of one number is 7 while value of the other is 4.
Ques: Ritu buys postage stamps of 25 paise and 50 paise worth Rs 10. She buys a total of 28 stamps. Find the total number of 25 paise stamps bought by Ritu? [5 marks]
Solution: According to the problem, number of 25 paise stamps is to be determined. Thus 25 paise and 50 paise stamps are unknown quantities and to be represented using variables. Since 25 paise and 50 paise stamps make a total of 28 stamps, let us consider number of 25 paise stamps to be ‘x’ and number of 50 paise stamps as ‘y’.
According to the problem statement,
x + y = 28 ------------------ (i)
25x + 50y = 1000 [Since Rs 10 makes 1000 paise]
x + 2y = 40 ------------------ (ii)
Subtracting (i) from (ii), we get
x + 12 = 28
x = 16
This implies the number of 25 paise coins is 16.
Ques. How can a linear equation be classified? [4 marks]
Ans: Linear Equation can be classified based on the number of variables available in the expression:
A linear equation with one variable is an equation with one variable and has a degree of one. 5x +30 = 0, 3x + 12 = 48 are examples of linear equation with one variable.
A linear equation with two variables is an equation with two variables and has a degree of one. 2x + 3y =12, 3x + 7y = 42 are examples of linear equation with two variables.
Ques. Illustrate the elimination method of solving linear equations? [3 marks]
Ans: The elimination method for solving linear equations involves multiplying one equation with a constant and then adding or subtracting all the equations to get an equation of one variable. These equations are then solved by the process of substitution to get the final solution.
Ques. How can you graphically represent a linear equation? [3 marks]
Ans: A linear equation can be represented graphically using the following equation.
ax + by + c = 0 where ‘x’ and ‘y’ are variables, ‘a’ and ‘b’ are coefficients and ‘c’ is a constant term.
Ques. How to calculate the slope of a line using a linear equation? [3 marks]
Ans: The equation for the slope of a line can be represented by y = mx + c where ‘x’ and ‘y’ are variables, c is the constant term, and ‘m’ is the slope of the line. Using trial and error methods, solutions of ‘x’ and ‘y’ points can be determined, using which the value of ‘m’ can be determined by substituting the values of ‘x’ and ‘y’ in the above equation.
Ques. Why a linear equation is called linear? [2 marks]
Ans: Since each term in a linear equation has an exponent of 1 and when expressed graphically it always results in a straight line, thus such equations are called a linear equation.
Ques. How many solutions does a linear equation have in two variables? [2 marks]
Ans: A linear equation in two variables can have infinitely many solutions. Since the graph of every linear equation with two variables can be represented in a straight line with every point on the line representing a solution of the linear equation.
Ques. State some real-life application of linear equation? [5 marks]
Ans: A wide range of real-life applications can be represented using linear equations, some of which are listed below:
- Finding solutions to age-related problems
- Determination of speed, distance, and time of a moving object.
- Finding solutions to geometry-related problems
- Problems based on money and percentage
- Finding solutions to force and pressure-related problems
- Problems based on work, time, and wages
Ques. What steps are to be followed for converting real-life situations into linear equations? [3 marks]
Ans: Following steps need to be followed for converting real-life situations into the linear equations:
- Convert the real-life problem into a mathematical statement to form an algebraic expression.
- Identify the unknown in such expressions and assign variables to such unknown quantities.
- Organize the obtained information using data, phrases by reading such mathematical statements multiple numbers of times.
- Represent such statements using algebra expressions based on the provided data and solve using systematic equations.
- Reframe the solution to the problem statement and verify whether that the problem.
Ques. What are the different methods of solving linear equations? [3 marks]
Ans: Linear equations can be solved using the following methods:
- Solving Linear Equations by Substitution Method
- Solving Linear Equations by Elimination Method
- Solving Linear Equations by Graphical Method
- Solving Linear Equations by Cross Multiplication Method
Ques. Mention some standard formulas used to solve word problems using linear equations? [2 marks]
Ans: Listed below are some of the standard formulas used to solve word problems using linear equations:
- Distance = Speed x Time
- Work Done = Rate of Work x Time to complete the work
- Amount of liquid in water = Percentage of solution x Volume of solution
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