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Chapter 3 – Pair Of Linear Equations In Two Variables

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

One of the most important sections of the class 10th board exams is the topic ‘Linear Equations.’ Here you will learn to use two or more variables like x, y, z, along with the constants to define any function. Do you remember the definition of functions and equations? Read on!

The pair of linear equations in two variables is a major topic in the class 10th NCERT syllabus and comes as a major algebraic topic in all exams over India. Let us recall the definition of equations as any mathematical expression having one or more variables in it. According to this definition, you can easily deduce the meaning of the pair of linear equations in two variables. You might have come across this, as it is a mathematical expression containing two variables in it. One variable is independent, and the other depends on the first one. Any equation of the general form ax + by + c = 0 is a two-variable linear equation. Let’s check on the various methods of solving any pair of given linear equations.

In the above equation of ax + by + c = 0, the x and y are variables, or we can call them arguments. And the a, b, and c are the constants. You can easily graph this equation on graph paper, which would ultimately form a line. Let’s check how.

In this method, we are given a pair of linear equations for which we need to find the solutions. The best way is to turn both the equations into dependent function forms, and then plot their graphs.

Example: You are given two linear equations, namely 

  • 2x + 2y = 3
  • 2x – 2y =5

Now both these equations can be written as:

  • x = (3 – 2y) / 2
  • x= (5 + 2y) / 2, respectively.

Now simply, mark all the points of x and y on the graph and form two lines. The point where the lines intersect each other is the solution to the given question.

In this method, we are given a pair of linear equations for which we need to find the solutions. We convert both the equations into the dependant function form by bringing one variable to one side of the = sign. And then we compare them. The solutions can be found in four possible ways, as depicted below.

  1. Substitution Method:

Example: You are given two linear equations, namely 

  • 2x + 2y = 3
  • 2x – 2y =5

Now both these equations can be written as

  • x = (3 – 2y)/2
  • x= (5 + 2y)/2, respectively.

Now, you take the value of x from the first equation and put it in the second equation in the place of x. Then after solving it, you get the value of y. Put that value in any one of the equations to get the value of x.

  1. Elimination Method:

Example: You are given two linear equations, namely 

  • 2x + 2y = 3
  • 2x – 2y = 5

You now add or subtract both the equations in such a way that any one variable gets cancelled. In the above example, you can simply add both equations, which would result in:

-4x = 8

-x = 2

Now put the value of x in any equation to get the value of y.

  1. Cross Multiplication Method:

This method follows a simple formula, which is given as

You have two linear equations

  • a1x + b1y + c1=0
  • a2x + b2y + c2=0,

You can calculate x and y with this formula.

  • x = (b1c2−b2c1) / (a1b2−a2b1
  • y = (c1a2−c2a1) / (a1b2−a2b1)
  1. Equations Reducible To A Pair Of Linear Equations In Two Variables:

All the above cases were for the linear equations, where the two variables can easily be checked. For some other examples, like, 2/x + 3/y = 5, we can simply assume 1/x = u and 1/y= w and change the given equation into something like this: 2u + 3w = 5. This is particularly for equations Reducible To A Pair Of Linear Equations In Two Variables.

After we have calculated u and w, we can easily put the real equations in place of u and w, then find the values of x and y.

Linear equation in two variables is an easy topic that is asked in the class 10th boards exam. There are various ways of solving a pair of linear equations, such as substitution method, graphical method, cross-multiplication method, and more. Choose one which is suitable for the question and solves it in less time.

  1. What is a pair of linear equations?

A linear equation is an equation in which all the operations between variables and constants are linear. For example,

a.x + b.y + c = d

+ and – are linear operations; they do not span dimensions and work on the same dimension.

* and / are different.

  • The equations are linear when one of the operants remains fixed, while the other part is variable. You can assume them to be repeated additions.
  • We call them non-linear when both the operands are not fixed, and are variables.
  1. How many solutions has a linear equation in two variables?

If the lines turn out to be ax + by = c and ax – by = d or ax + by = c and dx (+/-) ey = f, then you have two solutions. If you get any value as zero, it is still a well-defined solution.

If both lines are ax + by = c, then you have one solution, that is the whole line. You will not get a well defined solution.

If lines are ax + by = c and ax + by = d, then you have no solutions, as these two lines are parallel. Distance between the two lines will be |c-d| / √(a^2+b^2).

  1. How do you solve two equations with two variables?

There are many possibilities for solving two variable equations. 

  • In the elimination method, make one of the variables of both the equations the same, by dividing/ multiplying one or both the equations with a common factor. Repeat the same for the other variable. Or, substitute the value of the first equation in one of the original equations to find the second variable.
  • In the substitution method, write one of the equations in terms of the other variable, and substitute it into the other equation, then solve for the variable. Now substitute the determined variable in one of the original equations to find the other variable.
  1. What is a linear equation? Give 5 examples.

An equation is a certain kind of relation between an indeterminate and one or two constants (with respect to that indeterminate). By custom, the indeterminate is often symbolised as x or a, and either numeric constants or other alphabetic characters are used for the constants such as x, y, z, or a, b, c, and so on. It’s a relation that involves equality. In its simplest form, we have something like: ax + by = c

Some more examples are given below:

  • 2x – 34y = 33
  • 5x + 3f = 8
  • 9a – 3.2c = 0
  • 3a = b
  • 65d + 84s = 3
  1. How do you graph a linear equation in two variables?

The graph is an intersection between the two poles at 90 degrees. On one side, you can show the dependent function (y), and on the other side, there is the independent argument (x).

You can get more such questions for practice on MSVgo. There are some formulas of Pair Of Linear Equations In Two Variables that are critical and you can check out videos on MSVgo to understand the concept behind them.

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