Pair of Linear Equations in Two Variables

Consistency of Pair of Linear Equations in Two Variables

A pair of linear equations in two variables have the same set of variables across both the equations. These equations are solved simultaneously to arrive at a solution. In this article, we will look at the various types of solutions of equations in two variables.

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Types of Linear Equations in Two Variables

Solutions of linear equations in two variables can be of three types

  1. Single Solution
  2. Infinite Solutions
  3. No Solution

Understanding these types will help us in solving linear equations in two variables effectively. We will look at each of them in details.

Type 1: A single solution of a pair of linear equations in two variables

Consider the following pair of linear equations in two variables,

  • x – 2y = 0
  • 3x + 4y = 20

The solution of this pair would be a pair (x, y). Let’s find the solution, geometrically. The tables for these equations are:

x 0 2
y = (1/2)x 0 1

 

x 0 4
y = (20 – 3x)/4 5 2

Now, take a graph paper and plot the following points:

  • A(0, 0)
  • B(2, 1)
  • P(0, 5)
  • Q(4, 2)

Next, draw the lines AB and PQ as shown below.

equations in two variables

From the figure above, you can see that the two lines intersect at the point Q (4, 2). Therefore, point Q lies on the lines represented by both the equations, x – 2y = 0 and 3x + 4y = 20. Hence, (4, 2) is the solution of this pair of equations in two variables. Let’s verify it algebraically:

  • x – 2y = 4 – 2(2) = 4 – 4 = 0 = RHS
  • 3x + 4y = 3(4) + 4(2) = 12 + 8 = 20 = RHS

Further, from the graph, you can see that point Q is the only common point between the two lines. Hence, this pair of equations has a single solution.

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Type 2: Infinite solutions of a pair of linear equations in two variables

Consider the following pair of linear equations in two variables,

  • 2x + 3y = 9
  • 4x + 6y = 18

The solution of this pair would be a pair (x, y). Let’s find the solution, geometrically. The tables for these equations are:

x 0 4.5
y = (9 – 2x)/3 3 0

 

x 0 3
y = (18 – 4x)/6 3 1

Now, take a graph paper and plot the following points:

  • A(0, 3)
  • B(4.5, 0)
  • P(3, 1)

Next, draw the lines AB and AP as shown below.

equations in two variables

From the figure above, you can see that the two lines coincide. Therefore, there is no point of intersection between these two lines. Every point on the line represented by 2x + 3y = 9 is present on the line represented by 4x + 6y = 18. Hence, this pair of equations has an infinite number of solutions.

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Pair of Linear Equations in Two Variables

Type 3: No solution of a pair of linear equations in two variables

Consider the following pair of linear equations in two variables,

  • x + 2y = 4
  • 2x + 4y = 12

The solution of this pair would be a pair (x, y). Let’s find the solution, geometrically. The tables for these equations are:

x 0 4
y = (4 – x)/2 2 0

 

x 0 6
y = (12 – 2x)/4 3 0

Now, take a graph paper and plot the following points:

  • A(0, 2)
  • B(4, 0)
  • P(0, 3)
  • Q(6, 0)

Next, draw the lines AB and PQ as shown below.

equations in two variables

From the figure above, you can see that the two lines are parallel to each other. Therefore, these lines don’t intersect at all. Hence, this pair of equations has no solution.

Consistency of a Pair of Linear Equations in Two Variables

From the three examples above, we define the following terms

  • Inconsistent pair of linear equations: A pair of linear equations which has no solution. (Type 3 explained above)
  • Consistent pair of linear equations: A pair of linear equations which has a solution. (Type 1 and 2 explained above)
    • Independent pair of linear equations – If the pair of equations has only one solution.
    • Dependent pair of linear equations – If the pair of equations has infinite solutions.

To summarise, the lines represented by

  • (x – 2y = 0) and (3x + 4y = 20) intersect each other.
  • 2x + 3y = 9 and 4x + 6y = 18 coincide with each other.
  • ‘x + 2y = 4’ and ‘2x + 4y = 12’ are parallel to each other.

In General Form

If a1, b1, c1, a2, b2 and c2 are the coefficients of the equations in general form, then we can write the following table by comparing the values of a1/a2, b1/b2, and c1/c2 in all three examples.

Pair of lines a1/a2 b1/b2 c1/c2 Comparison Graphical Representation Algebraic Interpretation
x – 2y = 0 1/3 – 2/4 0/20 (a1/a2) ≠ (b1/b2) Intersecting lines One solution
3x + 4y – 20 = 0
2x + 3y – 9 = 0 2/4 3/6 9/18 (a1/a2) = (b1/b2) = (c1/c2) Coincident lines Infinite solutions
4x + 6y – 18 = 0
x + 2y – 4 = 0 1/2 2/4 4/12 (a1/a2) = (b1/b2) ≠ (c1/c2) Parallel lines No solution
2x + 4y – 12 = 0

From the table above, you can observe that if the lines represented by the equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are,

  1. Intersecting, then (a1/a2) ≠ (b1/b2)
  2. Coincident, then (a1/a2) = (b1/b2) = (c1/c2)
  3. Parallel, then (a1/a2) = (b1/b2) ≠ (c1/c2)

It is important to note that the converse is also true.

Solved Examples for You

Question 1: Check graphically whether the pair of equations ‘x + 3y = 6’ and ‘2x – 3y = 12’ is consistent. If so, solve them graphically.

Answer : Let us draw the graphs of both the equations. The tables for these equations are:

x 0 6
y = (6 – x)/3 2 0

 

x 0 3
y = (2x – 12)/3 – 4 – 2

Now, take a graph paper and plot the following points:

  • A (0, 2)
  • B (6, 0)
  • P (0, – 4)
  • Q (3, – 2)

Next, draw the lines AB and PQ as shown below.

equations in two variables

From the figure above, you can see that the two lines intersect at the point B (6, 0). Therefore, x = 6 and y = 0 is the solution of this pair of equations in two variables. Hence, it is Consistent.

Question 2:  Explain how can you solve an equation with two variables?

Answer: The procedure for solving systems of algebraic equations containing two variables involves beginning by moving the variables to the equation’s different sides. Afterwards, there should be a division of both sides of the equation by one of the variables in order to accomplish the solution for that variable. Afterwards, there should be plugging of that number into the formula.

Question 3: Explain two variable equation?

Answer: A linear system of two equations that consist of two variables refers to any system that a person can write in the form of ax + by = p and cx + dy = q.

Question 4: Explain the way of solving the equations system?

Answer: One can solve equations system with the help of following steps:

  • Solve for one of the existing variables. First, solve for y.
  • Place that equation and solve for x.
  • Substitute x = 4 x = 4 x=4 and finally solve for y.

Question 5: Explain what you understand by a nonlinear equation?

Answer: A system of nonlinear equations refers to a system of multiple equations in multiple variables that have at least one equation that does not happen to be linear.

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