Application of Linear Equations
What is a Linear Equation?
An equation is when two algebraic expressions are equated to each other using an equal “=” sign. An equation with a degree equal to one (the highest power of the variable in the equation is one) is termed a Linear Equation.
What is a Linear Equation in One Variable?
The equations having degree one and a single variable are called a linear equation in one variable.
For example:
4x=12
19x + 45 = 68
What is a Linear Equation in Two Variables?
The equations having degree one and two variables are called a linear equation in two variables.
For example:
3x + 5y = 15
18x = 17 – 2y
Representation of Linear Equations
The linear equation in two variables can be represented graphically. The (x, y) points on the graph are the solution set for the equation which makes the expressions match on both sides of the equal “=” sign.
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This is the graphical representation of a linear equation ax + by + c = 0. Here,

a and b are coefficients.

x and y are variables.

c is a constant term.
Applications of Linear Equations
There are various reallife examples of linear equations. These reallife problems are converted into mathematical forms to form linear equations which are then solved using various methods. It should clearly explain the relationship between the data and the unknowns (variables) in the situation. Below mentioned are the steps to be taken while converting a reallife problem into a linear equation:

writing the word problem as a mathematical statement in the form of an algebraic expression.

The quantities whose value can keep changing with time and different inputs are said to be variable quantities. These should be identified and assigned as variables.

The information given in the problem should be translated and written in a sequential manner.

After that, equations need to be framed with algebraic expressions and data cited in the word problems.

These linear equations can then be solved to find out the value of the unknown variables using various methods of equation solving.

The solutions should be retraced and verified for their correctness and to ensure that they meet all the criteria mentioned in the problem
Some Common Applications of Linear Equations in Real Life Involve Calculations of:

Age problems

Speed, time and distance problems

Geometry problems

Money and percentage of problems

Wages and hourly rate problems

Force and pressure problems
Solved Examples:
1) The Sum of Two Numbers is 60. One Number is Twice another Number. Find the Numbers.
Solution:
Let the two numbers be x and y.
According to the question, the sum of both numbers is 60.
Therefore, x + y = 60 ————————– (1)
Now, one number is twice the other number,
Considering y as the bigger number we can write 2x=y ————————– (2)
We will now use the substitution method of solving the solutions. Substituting y =
2x from (2) into (1) we get,
x + 2x = 60
so, 3x = 60
x = 60/3
x = 20
now since y = 2x
the value of y is 20 x 2 = 40
y = 40
verifying, x + y = 60
20 + 40 = 60
Since, LHS = RHS out solution is correct.
Answer) Therefore, the Two Numbers are 20 and 60.
2) If a car travels from city A to city B in 4 hours and covers a distance of 600 kms, find the speed of the car.
Solution:
Let the speed of the car be “S”, time be “t”, and distance be “d”.
We are aware of the relationship between speed, distance and time.
Speed = distance/time
S = d / t
Here d = 600, t = 4.
So, our equation becomes,
S = 600/4
S = 150
Hence, the speed of the car is 150 km/hr.
3) A daughter is onefourth her father’s age at present. If after 5 years, the daughter becomes onethird of her father’s age. Then, calculate their present ages.
Solution:
Let us take x and y as the present ages of the father and daughter.
Now, at present daughter is one fourth her father’s age
So, y = x/4 or x= 4y ————————(1)
After 5 years, so we add 5 to the present ages.
(y + 5) = 1/3 (x + 5)
Or, 3y + 15 = x + 5 ————————(2)
Substituting x = 4y from equation (1) to equation (2) we get,
3y + 15 = 4y + 5
4y – 3y = 15 – 5
Hence, y = 10
Now, x = 4y
So, x = 4 x 10 = 40.
Hence, x = 40.
Hence the present age of the father is 40, and that of the daughter is 10.
Do you know
1. Linear Equations were invented by the famous Irish mathematician, Sir William Rowan Hamilton in the year 1843.
2. Linear algebra enjoys a close relationship with linear equations. Linear algebra highlights linear equations and the relationship between variables.
1. How can we Calculate the Slope of a Line using Linear Equations?
Ans: The equation for the slope of a line is y = mx + c. here, x and y are the variables, c is the constant term, and “m” is the slope of the line. Using trial and error methods, we can find solutions for x and y points and then we will have only one unknown in our equation which is “m”, and we can find that by substituting the x and y values.
2. What is the Elimination Method of Solving Linear Equations and when can it be Used?
Ans: The elimination method is used when you have to solve a system of two linear equations which have more than one variable.
It involves multiplying one equation by a constant and then adding or subtracting the equations in order to get an equation in one variable. This can then be solved by substitution of values to get the final solution.