What is a Linear Equation in Two Variables?

A linear equation in two variables is an equation that can be written in the form:

ax+by=c

Here, a,b and c are constants, and x and y are the variables.
The solution to this equation consists of pairs of values for

x

and

y

that satisfy the equation.

For example, consider the equation: $2 x + 3 y = 6$

In this case,

a = 2

,

b = 3, and

c = 6

. If we substitute different values for

x, we can calculate the corresponding

y values..

Graphical Representation

A linear equation in two variables, when plotted on a graph, results in a straight line. Each point on the line represents a solution to the equation.

For example, The equation $2 x - 3 y = 6$ represents a straight line with infinitely many points along it. Here are a few points that lie on this line, plotted for reference.

Pair of Linear equations in 2 variables

This lesson on "Linear Equations in Two Variables" provides a comprehensive exploration of different methods to solve linear equations. It covers

(i) Graphical method

(ii) Elimination method

(iii) Substitution method

(iv) Cross multiplication method

Students will learn how to form and solve equations related to various scenarios, such as determining fractions, solving age-related problems, and tackling upstream-downstream problems in rivers.

Graphical method:

We have a pair of lines: a₁ x+b₁ y+c=0 and a₂ x+b₂ y+c=0

The graphical method is a visual way to solve a system of two-variable linear equations. The solution to the system is the point where the graphs of the two equations intersect. Let’s go through the steps to solve two equations graphically.

Example: Solve 2x+3y=8 and -3x+4y=5

To draw a line we need 2 points to plot and join them through a line.

First equation: 2x+3y=8

Let's take y=0 ⇒ 2x+3(0)=8 ⇒ x=4 ⇒ ( 4,0)

Let's take y=2 ⇒ 2x+3(2)=8 ⇒ x=1 ⇒ ( 1,2)

We can plot the points and join them through a line.

-3x+4y=5
Let's take x=1 ⇒ -3(1)+4y=5 ⇒ y=2 ⇒ ( 1,2)

Let's take x=-3 ⇒ -3(-3)+4y=5 ⇒ y=-1 ⇒ (-3,-1)

Again we plot the points and join them through a line.

The point where the two lines intersect i.e. ( 1,2) is the solution to the system of equations. This point gives the values of x and y that satisfy both equations.

Elimination method:
To solve a system of equations by elimination we use multiplication to transform the system such that one variable "cancels out" on adding them. So, we need to obtain the coefficients that are equal but opposite in sign.

Our GOAL is to obtain the coefficients that are equal but opposite in sign.

3x-2y=8 ............................. (1)

2x+5y=-1 .......................... (2)

Step 1: Let's make the coefficients of y equal.
Multiply eq(1) by 5 and eq(2) by 2. On adding 'y' cancels out.

( 3x-2y=8 ) X 5 ⇒ 15x - 10y = 40

( 2x+5y=-1 ) X 2 ⇒ 4x + 10y = -2

---------------

19x = 38

⇒ x = 2

Step 2: Plug in the value of x in eq(1) { we can choose any equation }

3x-2y=8

⇒ 3(2) - 2y=8

⇒ y=-1

Answer: (2,-1)

Substitution method:

In substitution method, we use one equation out of the two given. We get the value of one variable in term of the other variable i.e. we get ‘x’ in terms of y or we get ‘y’ in terms of x.

After that, we plug in this value in the second equation and simplify to get the value of that variable.

2x-y=8 .......................... (1)

3x+2y=5 ............................(2)
Using eq(1) we get y=2x-8 and plug this value into eq(2)
{ Generally we avoid the fractions to make the calculations easier. Keeping this in mind we should choose the equation and the variable. }
3x+2y=5
⇒ 3x+2(2x-8)=5
⇒ 7x=21
⇒ x=3
Use this value of x and plug this value into y=2x-8.
y = 2(3)-8=-2
Answer: (3,-2)

Cross Multiplication method:
We have 2 lines: a₁ x+b₁y+c₁ =0
and a₂ x+b₂y+c₂ =0
We setup the values in the order b, c, a, b below x, y , 1
The arrows between the two numbers indicate that they are to be multiplied and the second product is to be subtracted from the first.

Example: Solve: 2x+3y+2=0

and 3x-5y+22=0

Answer: (-4,2)

Free course ( Linear equations in two variables )

In addition to these methods, we delve into the process of forming and solving equations derived from word problems.

The course also addresses the concept of consistency of equations. We provide comprehensive examples that illustrate the different types of solutions—unique, infinite, or none—ensuring you grasp the conditions under which each occurs.

To reinforce learning, the course is enriched with video solutions that break down complex problems into manageable steps. You'll also have access to a variety of practice problems designed to test your understanding and help you master these critical concepts.

Frequently Asked Questions (FAQ)

1. What is the significance of solving linear equations in two variables?

2. How are linear equations graphically represented in a Cartesian coordinate system?

3. Can linear equations in two variables be used to solve real-world problems?

4. Are there different methods for solving systems of linear equations in two variables?

Hem Chandra Bhatt
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