Pair of Linear Equations in Two Variables Class 10

This chapter in Class 10 Syllabus about Pair of Linear Equations in Two Variables help students understand how to solve a problem. Maths is a topic that takes a lot of learning. The section also helps students in understanding that perhaps the answer to this problem is a set of values, one for x and another for y, which renders the two sides of both the equations identical. Students also discover that an answer to the equation is a location on the path that defines it. Students study the problem-solving approach of the problems posed in Chapter 3 Pair of Linear Equations in Two Variables, which revolves around the principles described below.

Basic Definitions

Before knowing how to solve Pair of Linear Equations in Two Variables in class 10, it is essential to know the following definitions:

Definition of an Equation

An equation is a statement that two mathematical expressions having one or more variables are equal.

Definition of an Linear Equation

• Equations in which the powers of all the variables involved are one are called linear equations. 
• The degree of a linear equation is always one.

General type of linear equations are represented in two variables where, 

The general form of a linear equation in two variables is ax + by + c = 0, whereby a and b cannot be zero at the same time. 

Describing linear equations for conceptual understanding is done by describing a word problem as a linear equation for a Pair of Linear Equations in Two Variables,

• Recognize and denote unknown quantities by variables. 
• Represents the relationship between variables in a statistical manner, replacing unknowns with variables.

Other Definitions

• For any linear equation, each solution (x, y) corresponds to a point on the line. General form is given by ax + by + c = 0.
• The graph of a linear equation is a straight line.
• Two linear equations in the same two variables are called a pair of linear equations in two variables. The most general form of a pair of linear equations is: a1x + b1y + c1 = 0; a2x + b2y + c2 = 0
  where a1, a2, b1, b2, c1 and c2 are real numbers, such that a12 + b12 ≠ 0, a22 + b22 ≠ 0.
• A pair of values of variables ‘x‘ and ‘y’ which satisfy both the equations in the given system of equations is said to be a solution of the simultaneous pair of linear equations.

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Two Types of Representation of Linear Equations For A Pair of Linear Equations in Two Variables

A pair of linear equations in two variables can be represented and solved, by using two methods which are:

1. Graphical method
2. Algebraic method

Plotting a Straight Line

The graph of a linear equation in two variables is a straight line. We plot the straight line as follows:

Any additional points plotted in this manner will lie on the same line.

General form of a pair of linear equations in 2 variables

A pair of linear equations in two variables can be represented as follows

The coefficients of x and y cannot be zero simultaneously for an equation.

Nature of 2 Straight Lines in a Plane for a Pair of Linear Equations in Two Variables

For a pair of straight lines on a plane, there are three possibilities which are:

i) They Intersect at Exactly One Point

Eg. Pair of Linear Equations which Intersect at a Single Point.

ii) They are Parallel

Eg. Pair of Linear Equations which are Parallel.

iii) They are Coincident

Eg. Pair of Linear Equations which are Coincident.

Solution for Consistent Pair of Linear Equations in Two Variables

The solution of a pair of linear equations is of the form (x,y) which satisfies both the equations simultaneously. Solution for a consistent pair of linear equations can be found out using

• Elimination method
• Substitution Method
• Cross-multiplication method
• Graphical method

Algebraic Method of Solution

Consider the following system of equation

a1x + b1y + c1 =0; a2x + b2y + c1 =0

There are following three methods under Algebraic method to solve the above system.

For the Pair of Linear Equations in Two Variables:

a1x + b1y + C1 = 0

a2x + b2y + c2 = 0

Consider the following diagram.

Solve it to get the solution, provided a1b2 – a2b1 ≠ 0

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Equations Reducible to a Pair of Linear Equations in Two Variables

Sometimes pairs of equations are not linear (or not in standard form), then they are altered so that they reduce to a pair of linear equations in standard form.

For example;

Here we substitute x = p & \frac{1}{y} = q, the above equations reduces to:

a1p + b1q = c1 ; a2p – b2q = c2

Solutions to Problems

1. Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be”. Isn’t this interesting? Represent this situation algebraically and graphically.

Solution:

To describe a given condition arithmetically, see what we really need to figure out about the question. Here, Aftab and his daughter’s existing age must be found in such a manner that the ages are expressed by the parameters x and y. The issue is about their ages 7 years ago and three years from now. Here, ‘seven years ago’ means that we have to deduct 7 from their existing year, and ‘three years from now’ means that we have to add 3 to their current year.

In order to graphically depict algebraic equations, the answer set of equations must be taken as whole numbers only for precision. The graph of the two linear equations is defined by a straight line.

2. The cost of 2 kg of apples and 1 kg of grapes on a day was found to be ₹160.  After a month, the cost of 4 kg of apples and 2 kg of grapes is ₹300. Represent the situation algebraically and geometrically.

Solution:

The cost of apples and grapes must be calculated so that the cost of 1 kg of apples and 1 kg of grapes is taken as factors. A pair of linear equations in two variables can be derived from the specified conditions of the combined expense of apples and grapes. of Then, in order to visually reflect the received equations, take the values of the variables as a whole of the numbers. Since these values are high, take the acceptable scale as 1 cm = 20.

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