Linear Equation in Two Variables

Class 9 Maths syllabus imparts students with the basics and foundational concepts of Maths. It includes a variety of integral mathematical concepts which are essential to understand the advanced-level topics you will get to study in Class 11 and Class 12. One such important topic is Linear Equations of Two Variables which elucidates upon the formation of linear equations when two variables are given and how it is different from the equation with one variable. This blog brings you a detailed study guide and notes on this topic to help you understand it in a simple manner.

Understanding Linear Equations in Two Variables

If a, b, and c are real numbers wherein (ax + by + c = 0) or (ax + by = c), and if a and b are not equal to 0 then the equation is said to be a linear equation having two variables. E.g. 10x – 3y = 5, 10x + 4y = 2, etc. 

a,b and c are real numbers and coefficients of x and y such that the solution for such an equation is a pair of values, each for x and y to make both the sides of an equation equal.     

An equation containing two variables has infinitely many solutions, but to find a solution, it is necessarily required to find both the equations. This chapter for class 9 teaches the core concepts to find these solutions for two-variable equations.

Linear Equation in Two Variable Formula

If a, b, and r are all real integers that are not equal to zero, then ax + by = r is a two-variable linear equation. These equations in two variables are indicated by x and y. The coefficients are represented by the integers a and b.

• The usual form of a two-variable linear equation is ax+by = r.
• In the preceding equation, the number ‘r’ is referred to as the constant.
• As seen in the following example, a linear equation with two variables contains three entities.
• 20x – 13y = 5 and 21x + 41y = 7 are examples of two-variable linear equations.

Solving Linear Equations in Two Variables

When solving a linear equation with two variables, the following principles must always be followed. The solution of a linear equation is not altered when: 

• The same number is added to (or subtracted from) L.H.S. and R.H.S. of the equation.
• Dividing or multiplying both L.H.S. and R.H.S. by the same non-zero integer.

Every linear equation with a single variable has a distinct solution. However, a pair of linear equations have two solutions, one for x and one for y, which satisfy both equations.

Unique Solution 

If a given set of linear equations crosses at a point, the solution for both equations will be unique. The slopes of the lines should differ from each other in order to provide a unique solution for a set of equations. Assume m1 and m2 are the slopes of two lines with two variables in their equations. In such a scenario, when m1 is not equal to m2, we may expect a unique answer or solution.

No Solution 

Lines will be parallel if the slopes of two two-variable equations are equivalent, i.e. m1 = m2. As a result of the absence of intersections, they will be unable to find a solution.

Unique and Infinite Solutions 

If a pair of linear equations is consistent, the lines will have both unique and infinite solutions, i.e. they will intersect or coincide.

For Example: 2x + 3y = 12 is an equation wherein the value of x=3 and y=2. When substituting the value of x and y in the given equation, it will make the L.H.S. equals to R.H.S.

Therefore, the solution will be written as an ordered pair (3, 2), writing the value of x first and then of y. Similarly, linear equations of two variables can have different solutions like (0, 4), (5,8), (2,4), etc. 

How To Represent Linear Equation in Two Variables on a Graph?

A linear equation of two variables can be easily represented algebraically as well as represented geometrically. An equation that is in the standard form of ax + by + c = 0 always has a pair of solutions (x, y). Hence, it can be easily represented through a coordinate plane on a graph. 

Thus, when we represent a linear equation on a graph geometrically through a line whose points make up the collection of solutions of the equation, then we can call it the graph of a linear equation. When it is placed graphically, there will be a straight line that may or may not cut the coordinate axes. 

In such a scenario, every solution for equations of having two variables will have a point on the line, and every point on the line will be a solution. If you want to obtain the graph of an equation, you have to mark two points corresponding to two solutions and hence, join them through a line. 

For Example: Let us consider the equation 6x + 3y = 12 and calculate the corresponding values and x and y by keeping each one of them 0. 

x axis	0	2
y axis	4	0

Important Questions and Solutions

You may think that this topic of mathematics is just limited to bookish knowledge and applicability. But the knowledge of this concept can be applied to solve reasoning questions which are commonly asked in competitive exams. Likewise, this concept has vast applicability that helps in the identification of unknown problems and solves them.   

1. Raju is twice as old as Rekha. 10 years ago his age was thrice of Rekha. Find their present ages. 

The age of Raju and Rekha are unknown. Assume the age of Rekha as ‘x’ years,
As per the question, the age of Raju will be two times that of Rekha, i.e. ‘2x’. 
Therefore, 10 years ago the age Rekha would be ‘x-10’, and that of Raju will be ‘2x-10’. 
As given, 10 years ago, Raju’s age was thrice of Rekha, i.e. 2x – 10 = 3(x – 10).
2x – 10 = 3 (x – 10)
2x – 10 = 3x – 30
x = 20  

2. Find the value of variables which satisfies the following equations: 8x + 7y = 38 and 3x – 5y = -1.

Using the method of substitution to solve the pair of linear equation, we have:

a. 8x + 7y = 38

b. 3x – 5y = -1

Multiplying equation (a.) by 5 and (b.) by 7, we will have:

c. 40x + 35y = 190

d. 21x – 35y = -7

Adding equation (c) and (d), we will get:

61x  = 183

Where, x = 3

Substituting the value of x in either equation (a.) or (b.), we will get:

3 x (3) – 5y = -1

9 – 5y = -1

10 = 5y

Y = 10/5 = 2

Therefore, x = 3 and y = 2 is the point where the given lines intersect.

3. Draw the graph of the following linear equation in two variables:  x + y = 4

Given Equation: x + y = 4

If we assume x=0, then y=4

If we assume y=0, then x=4

With this, we get the following table,

x	0	4
y	4	0

Plotting the ordered pairs (0, 4) and (4, 0) on the graph paper and joining these points, we get a straight line AB.

4. Draw the graph of the following linear equation in two variables:  x – y = 2

Given Equation: x – y = 2

If we assume x=0, then y= -2

If we assume y=0, then x=2

With this, we get the following table,

x	0	2
y	-2	0

Plotting the ordered pairs (0, -2) and (2, 0) on the graph paper and joining these points, we get a straight line PQ.

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Linear Equation in Two Variables: Practice Questions

If you want to getter a better hold of the topic, here are some important questions you can practice:

1. How is the linear equation in two variables different from the linear equation in one variable?
2. How are the fundamentals of linear equations applicable in solving real-life problems?
3. How to represent the linear equation solution on a number line?
4. Find the value of variables which satisfies the following equation: 2x + 5y = 20 and 3x+6y =12.
5.  A boat running downstream covers a distance of 40 km in 3 hours while for covering the same distance upstream, it takes 6 hours. What is the speed of the boat in still water?
6. A boat running upstream takes 6 hours 10 minutes to cover a certain distance, while it takes 3 hours to cover the same distance running downstream. What is the ratio between the speed of the boat and the speed of the water current, respectively?
7. The sum of the digits of a two-digit number is 8. When the digits are reversed, the number is increased by 18. Find the number.
8. Jake’s piggy bank has11 coins (only quarters or dimes) that have a value of $1.85. How many dimes and quarters does the piggy bank has?
9. In a river, a boat can travel 30 miles upstream in 2 hours. The same boat can travel 51 miles downstream in 3 hours. Then
   • What is the speed of the boat in still water? 
   • What is the speed of the current?
10. Draw the graph of the following linear equation in two variables: 3x + 2y = 6
11. Mrs. Ahuja Lost Her Wallet that had 50 Rupees and 200 Rupee Notes that Amount to 1800 in a Mall. Represent the Composition of the Wallet as an Equation and Draw the Graph.
12. Draw the Graph of the Equation 2x + 3y = 12 and Find the Coordinates of the Point:
    • Where Y-coordinate is 3
    • Where X-coordinate is −3
13. Give the equations of two lines passing through (2, 14). How many more such lines are there, and why?
14. If the point (3, 4) lies on the graph of the equation 3y = ax + 7, find the value of a.
15. The taxi fare In a city Is as follows: For the first kilometer, the fare Is Rs. 8 and for the subsequent distance it is Rs. 5 per km. Taking the distance covered as x km and total fare as Rs.y, write a linear equation for this Information, and draw Its graph.
16. If the work done by a body on the application of a constant force is directly proportional to the distance traveled by the body, express this in the form of an equation in two variables and draw the graph of the same by taking the constant force as 5 units. Also read from the graph the work done when the distance traveled by the body is
    • 2 units
    • 0 unit
17. Yamini and Fatima, two students of Class IX of a school, together contributed Rs. 100 towards the Prime Minister’s Relief Fund to help the earthquake victims. Write a linear equation that satisfies this data. (You may take their contributions as Rs.x and Rs.y.) Draw the graph of the same.
18. Give the geometric representations of 3x + 9 = 0 as an equation
    1. in one variable
    2. in two variables
19. For the following questions use the Method of Substitution to find the solution to the given system or to determine if the system is inconsistent or dependent.
    1. x – 7y = -11 and 5x + 2y = -18
    2. 7x – 8y = -12 and -4x + 2y = 3
    3. 3x + 9y + -6 and -4x – 12y = 8
20. For the following questions, use the Method of Elimination to find the solution to the given system or to determine if the system is inconsistent or dependent.
    1. 6x – 5y = 8 and -12x + 2y = 0
    2. -2x + 10y = 2 and 5x – 25y = 3
    3. 2x + 3y = 20 and 7x = 2y = 53

Linear Equation in Two Variables: NCERT PDF

Linear equations of two variables is a key concept of Mathematics included in Class 9 to familiarise students with the basics of algebra and equations. Need career guidance to actualize your academic and professional goals? Our Leverage Edu experts are just a click away from assisting you at every step of your educational and career journey! Sign up for a free e-meeting today!