An equation in the form of px + qy + r, where p, q and r are real numbers and the variables p and q are not equivalent to zero, is known as a linear equation in two variables.
The two linear equations with similar two variables are considered as the pair of linear equations in two variables. There are two methods to solve a pair of linear equations in two variables. These two methods are:
Graphical method
Algebraic Method
The general form of pair of linear equation in two variables is given as:
p1x + q1y+ r1 = 0 p2x + q2y + r2 = 0
Here, p1, q1, r1, p2, q2, r2 are real numbers
P12 + q12 ≠ 0
p22 + q22 ≠ 0
We have studied linear equations in one variable in our previous classes and we know how to solve the linear equation in one variable. Solving the linear equation in one variable is simple and can be calculated easily. In order to solve the linear equations in two variables, two different sets are needed which will be helpful to determine the values of two unknown variables say x and y. If one equation is given and two variables are asked to be solved, we will not be able to get a particular solution for a given equation.
For example
2x + 5y = 6 and 5x + y = 2
We can get a particular solution for the above two equations. But , on the other side
We cannot get a particular solution for the equation 3x + 6y = 5 as only one equation is given and we are asked to find the values of two unknown variables. Hence, we can express the equation 3x + 6y = 5 as, y = (5-3x)/6
The values of y will change accordingly on the basis of the values of variable x, Hence, the particular solution for a given equation is not possible.
It can be said that two distinct sets of independent conditions are required to obtain a particular solution of systems of linear equations in two variables
There are generally two methods to solve and express the pair of linear equations in two variables. These two methods are:
Graphical Method
Algebraic Method
The general representation of pair of linear equation for two variables say x and y is given by:
p₁x + q₁y + r₁ = 0………….(1)
p₂x + q₂y + r₂ = 0 …………(2)
Where , p₁ q₁, r₁, p₂, q₂, r₂ are real numbers and p₁² + q₁² ≠ 0 and p₂²+ q₂² ≠ 0.
If the pair of linear equations are given in the form of p₁x + q₁y + r₁ = 0 and p₂x + q₂y + r₂ = 0, then the following situation takes place.
If the given pair of linear equations is consistent then p₁/p₂ ≠ q₁/q₂
If the given pair of linear equations is inconsistent then p₁/p₂ = q₁/q₂ ≠ r₁/r₂
If the given pair of linear equations is dependent and consistent then p₁/p₂ = q₁/q₂ = r₁/r₂.
For example, the two equations such as 2x- y = -1 and 3x + 2y = 9 are pairs of linear equations with two variables x and y. The diagram given below represents the solutions for both the equations by substituting the value of x to determine the value of y.
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Similarly, we can find the solutions for other pairs of linear equations in two variables.
As we know, the graphical representation of linear equations in two variables always forms a straight line. Therefore, a pair of linear equations in two variables forms a straight line which are observed together. There are three possibilities when two lines are drawn in a plane. These three possibilities are :
Two lines intersect with each other at one point.
Two lines which are drawn are parallel to each other.
The two lines will coincide with each other.
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Now, let us take a pair of linear equations such as x + y + 5 and 2x + 2y = 10. To represent these pair of linear quotations in a graph, we need to determine the solutions.
We will get two different values of x and y after determining the solutions. Hence it can expressed as below:
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Pair of linear equations are found in every situation. Let us learn the algebraic method of solving a pair of linear equations of two variables by considering practical application of daily life. For example, suppose you went to the fruits market. There were two types of oranges available. The fruit seller said that the small oranges are 3 times smaller than the large oranges. The total money the lady was having with her is Rs.100. Can you determine how much the lady spent to purchase two types of oranges?
Let us understand the above situation mathematically,
Let us consider the price of the smaller oranges = x
And ,the price of the bigger oranges = y.
As per the first situation in the questions x = 4y…………(1)
And as per the second situation in the questions, x + y = 100……………(2)
For determining the solution, we are required to find the value of both x and y and solve both the equations. The coordinates (x,y) can be represented easily in a graph. But the graphical method is the most easier way in the situations when the point representing the solution of the linear equations includes non-integral coordinates like ( √2, 5√6 ) , (–1.42, 5.3), (6/19, 1/17), etc. Hence, the algebraic methods are used to solve such situations.
The two algebraic methods to solve a pair of linear equations:
Substitution Method
Elimination Method
Cross-Multiplication Method
1. The sum of 4 times a larger integer and 5 times a smaller integer is 7. When twice the smaller integer is subtracted from 3 times the larger, the result is 11.Find the integers.
Solution:
We will first assign the variables to the larger and smaller integer.
Let us consider x as the smaller integer and y as the larger integer.
When Using two variables, we are required to form two equations. The first sentence explains addition and the second statement explains the difference.
This gives the following situation
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This gives us the following two equations.
4x + 5y = 7
3x- 2y = 11
We will solve the above equation through the substitution method.
In the substitution method, we will multiply the equation 4x + 5y = 7 by -2 and the equation 3x - 2y = 11 by 5
Hence, two equations which we got after multiplication are given below:
-8x -10y = -14……(1)
15x - 10y = 55…..(2)
Now, we will subtract the equation (2) from the equation (1)
(15x - 10y = 55) - ( -8x -10y = -14)
23x = 69
x = 69/23
x = 3
Now, we will substitute the value of x to find the value of y
4x + 5y = 7
4(3) + 5y = 7
12 + 5y = 7
5y = -5
y = -1
Hence, the larger integer is 3 and the smaller integer is -1
2. Calculate the value of variables which meets the following equation
2x + 5y = 20 and 3x+6y =12.
Solution:
We will solve the pair of linear equations by substitution method .
Two equations are given below:
2x + 5y = 20…………………….(1)
3x + 6y =12……………………..(ii)
Now, we will multiply the equation (i) by 3 and equation (ii) by 2,
After multiplication, we get following two equations
6x + 15y = 60…………………….(iii)
6x+12y = 24……………………..(iv)
Now , we will Subtract the equation (iv) from (iii)
(6x+12y = 24) - (6x + 15y = 60) = 0
-3y = - 36
y = 12
By substituting the value of y in any of the equation (i) or (ii), we get the value of x
2x + 5(12) = 20
x = −20
Hence x=-20 and y =12 are two point where the given equations intersect
1. What are the different steps to write linear equations in two variables?
Ans. The different steps to write linear equations in two variable are as follows:
The first step is to define the variables.
The second step is to write any two equations that include any two variables.
The third step is to state the equation in slope intercept form i.e y = ax + b
The fourth step is to graph the equation with the help of the y-intercept and slope.
The fourth step is to recognise the variables by observing the points where the intercepts meet.
The sixth step is to mention what is meant by points with reference to the problems.
2. What is the difference between Linear equations and Nonlinear equations?
Ans. The difference between linear equation and non-linear equation is stated below in tabulated from:
Linear Equation | Non Linear Equation |
Linear equation forms a straight line or represent the equation for straight line | Nonlinear equation does not form straight line but forms curve |
Linear equation has only degree 1 | A nonlinear equation has degree 2 or more than degree 2 but not less than degree 2. |
Linear equations forms a straight line in XY-plane and can be lengthen in any direction but in straight form | Nonlinear equation forms curve and if the curve is been increased, the curvature of the graph increases |
Linear equation is expressed in the form of y = ax + b Here, x and y represent the variables, a is the slope of line and c is a constant value. | Nonlinear equation is expressed in the form of ax² + by + c Here, x and y represent the variables, a and c is the slope of the line and constant value respectively. |
Example 9x =1 3x + y + 2 =0 3y = 2x | Examples x² + y² = 2 3x² + x + 2= 5 |