Let us consider a simple example of a linear equation in one variable
(i) 2x = 8
Here, x = 8/2 = 4 (Unique solution)
If we plot the solution of this equation on the graph we get a straight line.
In this chapter, we recall our previous knowledge and extend it to that of the linear equation in two variables.
Linear Equations
An equation in which the maximum power of the variable is one is called a linear equation.
Example: x – 2 = 5, x + y = 15 and 3x – 3y = 5 are some linear equations.
Linear equations can be used to solve reallife problems such as
(i) To know the cost of five pencils if the cost of one pencil is known.
(ii) Weather predictions
(iii) To express cricket score
(iv) To know how many chocolates and balloons we can buy for the money we have
(v) Government surveys.
(1) Linear Equation in one variable: The equation of the form ax + b = 0, where a and b are real numbers such that a ≠ 0 and x is a variable, is called a linear equation in one variable.
Example:
(i) 3x + 3 = 12,
(ii) t + 2t = 7 – t
(2) Linear Equation in two variables: An equation of the form ax + by + c = 0, where a, b and c are real numbers, such that, (a and b are not both zero), and x and y are the two variables, is called as a linear equation in two variables.
Example:
(i) 3x + 2y − 5 = 0,
(ii) x − 4 = √3y
Let us consider another example of linear equation in two variables,
One day, Richa and Pranita went to Mango orchard. They both started collecting mangoes, after two hours they got tired and sat under a tree. After some time they started counting the number of mangoes collected and found that they have collected 79 mangoes in 2 hours. If we have to represent a situation in the form of an equation it is written as
x + y = 79
Here,
x: Number of mangoes collected by Richa
y: Number of mangoes collected by Pranita
We don’t know how many mangoes are collected by each one of them i.e., there are two unknown quantities. Hence, we used x and y to denote them.
So, x + y = 79
This is the required equation.
The solution of a linear equation is not affected when:
(i) The same number is added (or subtracted) from both sides of the equation.
Example: (i) 4 + 2 = 2 × 3
When we subtract 5 from both sides of the given equation we get;
(4 + 2) − 5 = (2 × 3) − 5
6 − 5 = 6 − 5
1 = 1
LHS = RHS
Hence, we can conclude that the solution of a linear equation is not affected when the same number is subtracted from both sides of the equation.
Example: (ii) 4 + 2 = 2 × 3
Added 5 to both sides of given equation and we get;
(4 + 2) + 5 = (2 × 3) + 5
6 + 5 = 6 + 5
11 = 11
LHS = RHS
Hence, we can conclude that the solution of a linear equation is not affected when the same number is added to both sides of the equation.
(ii) The same nonzero number is multiplied or divided both sides of the equation.
Example: (i) 4 + 2 = 9 − 3
Multiplied by 5 to the both sides of given equation and we get; (4 + 2) × 5 = (9 − 3) × 5
⇒ 6 × 5 = 6 × 5
30 = 30
LHS = RHS
Hence, we can conclude that the solution of a linear equation is not affected when we multiply both sides of the equation by the same nonzero number.
Example:(ii) 4 + 2 = 9 − 3
Divide by 3 to both sides of the given equation and we get;
(4 + 2) / 3 = (9  3) / 3
(6) / 3 = (6) / 3
2 = 2
LHS = RHS
Hence, we can conclude that the solution of a linear equation is not affected when we divide both sides of the equation by the same non zero number.
Let’s solve some examples on Linear equations:
Example: Write each of the following equations in the form ax + by + c = 0 and indicate the values of a, b and c in each case.
(i) 3x + 4y = 7,
(ii) 2x – 8 = √3y,
(iii) x = 4y,
(iv) X/2  Y/4 = 5
(i) 3x + 4y = 7
The above equation can be rewritten as
3x + 4y − 7 = 0
On comparing with ax + by + c = 0, we get
a = 3, b = 4, and c = −7
(ii) 2x – 8 = √3y
The above equation can be rewritten as
2x – √3y − 8 = 0
On comparing with ax + by + c = 0, we get
a = 2, b = √3 , and c = − 8
(iii) 2x = 4y
The above equation can be rewritten as
2x – 4y = 0
There is no constant value in the given equation. So, we can write the equation as
2x − 4y + 0 = 0 ........ [Adding zero does not change the equation]
On comparing with ax + by + c = 0, we get
a = 2, b = − 4 , and c = 0
(iv) X/2  Y/4 = 5
2X  Y / 4= 5
2x − y = 5 × 4
2x − y = 20
2x − y − 20 = 0
On comparing with ax + by + c = 0, we get
a = 2, b = −1 , and c = −20
Write each of the following as an equation in two variables x and y
(i) x = −5
(ii) 3x = 7
(iii) 7y = 2
(i) x = −5
x + 5 = 0
1. x + 0. y + 5 = 0
(ii) 3x = 7
3x − 7 = 0
3. x + 0. y − 7 = 0
(iii) 7y = 2
7y – 2 = 0
0. x + 7. y – 2 = 0
Example: The cost of one book is thrice the cost of a pencil. Write the linear equation in two variables to represent this statement.
Let the cost of a book be Rs. x and that of a pen to be Rs.y. Then according to the given statement, we have
x = 3y
[OR]
1. x – 3y + 0 = 0
Example: The bus fare is as follows: For the first kilometre the fare is Rs. 6 and for the subsequent distance it is Rs. 4 per km. Taking the distance covered as m km and total
fare is Rs. n. Write a linear equation for this information.
Total distance covered = m km
Fare for the first km = 6 Rs.
Fare for rest of the distance = Rs. (m  1)4
We already know the total fare given is n
Total fare = [6 + (m  1)4]
n = 6 + 4m – 4
n = 4m + 6  4
n = 4m + 2
4m – n + 2 = 0
This is the linear equation for this information.
Example: John and Jimmi are two friends they are studying in class 9. Together they contributed 8 dollars for flood victims. Write a linear equation that satisfies the given data. Also, draw a graph for that.
Let the amount that John and Jimmi contributed to be x and y respectively.
Amount contributed by John + Amount contributed by Jimmi = 8. So, x + y = 8
This is a linear equation that satisfies the data.
To draw a graph we need to find a x and y coordinate which satisfy the equation
x + y = 8 ..........(1)
Put y = 0 in the equation (1)
x + 0 = 8
x = 8
x + y = 8..........(1)
Put x = 0 in the equation (1)
0 + y = 8
y = 8
So, it can be observed that (8, 0), (0, 8) satisfy the equation 1.
Therefore these are the solution of the above equation.
The graph is constructed as follow:
(i) Solution of Linear Equations in One Variable
“Any value of the variable that satisfies the given equation in x is called a solution or roots of the equation. We know the Linear equation in one variable has a unique solution.
Example:
(i) 3x + 3 = 12
3x = 12 – 3
3x = 9
x = 9/3
x = 3 is a solution of the given equation, which is unique.
(ii) 2x – 7 = 0
i. e. , 2x = 7
x = 7/2 is a solution of the given equation, which is unique.
(ii) Solution of Linear Equations in two Variables
There are two variables in the equation, a solution means a pair of values, one for x and one for y which satisfy the given equation. A linear equation in two variables has infinitely many solutions.
Example:
(i) x + 3y = 5
LHS = x + 3y
If we put x = 2 and y = 1 in the LHS of the equation we get,
LHS = x + 2y = 3 + 3(1) = 5
Here, LHS = RHS
So, we can say that x = 2 and y = 1 is a solution of the equation x + 3y = 5
Let us consider one more value of x and y.
Putting x = 1 and y = 2 in the LHS of the equation we get,
LHS = x + 3y
LHS = x + 3y = 1 + 3(2) = 7
Now, LHS ≠ RHS
Therefore, we can say that x = 1 and y = 2 is not a solution of the equation x + 3y = 5.
(ii) Show that (x = 1, y = 1) is a solution of 3x + 2y − 5 = 0
If we put x = 1 and y = 1 in the given equation
We have,
LHS = 3(1) + 2(1) − 5 = 0
LHS = 3 + 2 − 5 = 0
LHS = 5 − 5 = 0 = RHS
So, x = 1, y = 1 is a solution of 3x + 2y − 5 = 0
Write Find the value of k in the following case, if x = 2, y = 1 is a solution of the equations:
(i) 3x + 2y = k
Put x = 2, y = 1,then
3(2) + 2(1) = k
6 + 2 = k
k = 8
(ii) + = 5k
Put x = 2, y = 1, then
k =
For what value of k, the linear equation 2x + ky = 6 has x = 2 and y = 1 as its solution? If x = 4, then find the value of y.
The linear equation is 2x + ky = 6
At x = 2 and y = 1
2(2) + k(1) = 6
4 + k = 6
k = 6 – 4
k = 2 ... ... ... . (1)
If x = 4,then
2x + ky = 6
2(4) + 2y = 6 ....... [k = 2 from (1)]
8 + 2y = 6
2y = 6 – 8
2y = −2
∴ y = −1
Consider the points (2, 0), (3, 0), (5, 0) etc. In the given points they coordinates are zero hence, they lie on the xaxis. Thus, the equation of xaxis in one variable is y = 0 and the equation of xaxis in two variables is 0.x + 1.y = 0. This equation shows that for each value of x, the corresponding value of y is zero.
The graphical representation of xaxis and yaxis as follow:
Equation of Line parallel to yaxis:
Let x – 2 = 0 be an equation. Then,
(i) If it is treated as an equation in one variable x only, then it has a unique solution i.e., x = 2
(ii) If it is treated as an equation in two variables x and y, then it can be written as 1. x + 0. y – 2 = 0. This equation has infinitely many solutions of the form (2, a), Where a is any real number. Also, every point of the form (2, a) is a solution of this equation. Thus, the given equation represents the equation represents a line parallel to the yaxis.
So, the equation in two variables x – 2 = 0 is represented by line AB as shown in the figure
Equation of Line parallel to xaxis:
Let y – 2 = 0 be an equation. Then,
(i) If it is treated as an equation in one variable y only, then it has a unique solution i.e., y = 2
(ii) If it is treated as an equation in two variables x and y, then it can be written as 0. x + 1. y – 2 = 0. This equation has infinitely many solutions of form (a, 2) where a, is any real number.
Also, every point of the form (a, 2) is a solution of this equation. Thus, the given equation represents the equation represents a line parallel to xaxis.
So, the equation in two variables y – 2 = 0 is represented by line AB as shown in the figure
Example: Solve the equation 2y – 3 = 12 – y. Represent the solution
(i) in the number line
(ii) in the cartesian plane
The given equation is 2y – 3 = 12 – y
2y + y = 12 + 3
3y = 15
y = 15/3
y = 5
(i) If we treated y = 5 as an equation in one variable only, then it has a unique solution y = 5. So, it is a point on the number line.
(ii) If we treated y = 5 as an equation in two variables, then it can be written as 0. x + 1. y = 5 it is represented by a line. Here all the values of x are always zero. Because 0. x = 0. However, y must satisfy the equation y = 5
1. What is a linear equation in two variables? 
2. How do you graph a linear equation in two variables? 
3. What is the solution to a system of linear equations in two variables? 
4. Can a system of linear equations in two variables have no solution? 
5. How can we solve a system of linear equations in two variables algebraically? 
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62 videos426 docs102 tests
