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Chapter 3 in Class 10 Mathematics, Pair of Linear Equations in Two Variables, is an important algebraic concept that involves solving linear equations with two unknowns in several different ways. It involves three such methods: graphical, substitution, and elimination methods—all of them relevant to where two lines may cross and determine their solution. In other words, it must be grasped not just in the academic context but for practical purposes of solving problems in business, physics, and engineering. Class 10 Maths Chapter 3 Notes covers the solution of linear equations, interpretation of graphical solutions, and the use of such techniques for solving word problems.
Practising questions from class 10 ch 3 math can help in scoring at least 3 marks in the final examination. On this platform, students may get pertinent study materials like class 10 maths chapter 3 notes PDF, experiential exercises, DoE worksheets, and assistance materials from Educart.
Class 10 math Chapter 3 notes cover all the main concepts like substitution method and solving linear equations. Pair of Linear Equations in Two Variables include every number other than the complex numbers and the downloadable notes PDFs provided below are detailed and in easy-to-understand language.
Students can attempt a Pair of Linear Equations in Two Variables question answer worksheets to prepare themselves as per the exam pattern. Below we have provided the links to downloadable PDFs of DoE Worksheets for class 10 Mathematics to practice more questions.
Below we have provided the links to downloadable PDFs of Experiential Learning Activity for class 10 Mathematics to help students implement their acquired knowledge in the real world.
Below we have provided the links to downloadable PDFs for class 10 Mathematics linear equations formulas to help students solve complex questions and understand the concepts easily.
Below we have provided the links to downloadable PDFs of mind maps for class 10 Mathematics to help students implement their acquired knowledge in the real world.
Below we have provided Class 10 Mathematics Important Questions that cover questions from the NCERT textbook.
Below we have provided Class 10 Mathematics Question Banks that cover every typology question with detailed explanations from various resources in one place.
Below we have provided Class 10 Mathematics Support Material that covers Case Study-based questions from the various concepts explained in Math NCERT chapters.
In algebra, a linear equation is an expression that has the greatest exponent that the variable carries. In the case of two variables, such as x and y, the general form of a linear equation is as follows:
ax + by = c
In this equation:
An example of a linear equation in two variables is 2x + 3y = 5.
A system of two linear equations in two unknowns is termed a pair of linear equations in two variables. For example,
a1x + b1y = c₁
a2x + b2y = c2
Where a₁, b₁, and c₁ are constants in equation 1 and a₂, b₂, and c₂ are constants in equation 2. Here, the problem is to find the values of x and y that simultaneously satisfy both given equations.
There are three ways to solve a system of linear equations:
The graphical method involves plotting the two equations on the Cartesian plane and finding the point of intersection at which both of them have intersected to get a solution to the system.
Following are the steps to solve a system of linear equations.
Example:
Solve the following system of equations:
x + y = 4
2x + y = 5
Graph the above equations to determine the values of x and y at either point of intersection.
In this problem, the solution is the point (1, 3), where x = 1 and y = 3.
In the substitution method, you first solve one equation for one variable in terms of the other, and then you substitute that expression into the second equation.
Steps:
Example:
Solve the system of equations:
3x + 4y = 10
x - y = 1
From the second equation, express x in terms of y:
x = y + 1
Now, substitute into the first equation:
3(y + 1) + 4y = 10
Solving,
3y + 3 + 4y = 10
7y = 7
y = 1
Substitute y = 1 into the expression x = y + 1:
x = 1 + 1 = 2
Therefore, x = 2 and y = 1 is the solution to the system.
In the elimination method, we change the equations by adding or subtracting so that we eliminate one variable and solve for the other remaining variable.
Steps:
Example:
Given the following system of equations, solve:
2x + 3y = 12
4x - 3y = 6
Adding the two equations, we get:
(2x + 3y) + (4x - 3y) = 12 + 6
6x = 18
x = 3
We now substitute x = 3 into the first equation:
2(3) + 3y = 12
6 + 3y = 12
3y = 6
y = 2
Thus the solution is x = 3 and y = 2.
There are three possible solutions for a pair of linear equations.
Unique Solution:
If there exists one point at which a system of two equations intersects, then there is a unique or exact solution to that system of equations. Therefore, it establishes that the two lines are different and not parallel.
No Solution:
If the graphs of the equations are parallel and never intersect, then no point and thus no solution exists. In other words, there is no solution if the two equations are inconsistent.
Infinite Solutions:
When the two graphs coincide, that is, they are the same graph; the system has infinitely many solutions. In that case, the equations are dependent.
Linear equations do not only contain theory; they have their applications in real life, too, in many fields. Some of the everyday situations in which two linear equations work are listed below:
Business and Economics: Solving systems of linear equations helps you calculate profit/loss, expenses, and trends of supply vs. demand when dealing with companies. For example, knowing how to use your cost and profit functions, one can determine break-even or pricing at which points profit is reached.
Travel Problems: Linear equations can be used to solve problems involving rate, time, and distance. Imagine two cars travelling toward each other from two different cities; you can use the system of linear equations to determine when and where they will meet.
Mixture Problems: When a variety of liquids are mixed (like different solutions of water and alcohol), linear equations tell how much of each liquid must be taken to produce a desired mixture.
Geometry and Physics: In geometry, linear equations tell where two lines or planes intersect. In physics, they represent relationships between different physical quantities.
In this chapter, you have learned how to deal with pairs of linear equations in two variables and several methods which might be applied in solving them. All of these techniques are important if you are going to overcome algebraic hurdles as well as real-life problems. Graphically or by substituting, or even by elimination, the key is to practice until mastery is gained in solving linear equations.
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