Equation: Assertion that two mathematical expressions with one or more variables are equal is called an equation.
Equation of a Line: Linear equations are those in which the powers of all the variables involved are equal. A linear equation's degree is always one.
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A Linear Equation in Two Variables in Its General Form
A linear equation in two variables has the generic form ax + by + c = 0, where a and b cannot be zero at the same time.
Students can use the short notes and MCQ questions, as well as the standalone notes.
For easy revision, get the solution pdf for this chapter from the links below:
For a word problem, representing linear equations: Using a linear equation to illustrate a word problem
CBSE Class 10 Maths Notes of All Chapters:
CBSE CLASS 10 MATHS NOTES CHAPTER 1 REAL NUMBERS
CBSE CLASS 10 MATHS NOTES CHAPTER 2 POLYNOMIALS
CBSE CLASS 10 MATHS NOTES CHAPTER 3 PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
CBSE CLASS 10 MATHS NOTES CHAPTER 4 QUADRATIC EQUATION
CBSE CLASS 10 MATHS NOTES CHAPTER 5 AIRTHMETIC PROGRESSION
CBSE CLASS 10 MATHS NOTES CHAPTER 6 TRIANGLE
CBSE CLASS 10 MATHS NOTES CHAPTER 7 CONTROL AND COORDINATION
CBSE Class 10 Maths Notes Chapter 8 Introduction To Trigonometry
CBSE CLASS 10 MATHS NOTES CHAPTER 9 SOME APPLICATIONS OF TRIGONOMETRY
CBSE Class 10 Maths Notes Chapter 10 Circle
CBSE CLASS 10 MATHS NOTES CHAPTER 11 CONSTRUCTION
CBSE CLASS 10 MATHS NOTES CHAPTER 12 AREAS RELATED TO CIRCLES
CBSE CLASS 10 MATHS NOTES CHAPTER 13 SURFACE AREA AND VOLUMES
CBSE CLASS 10 MATHS NOTES CHAPTER 14 STATISTICS
CBSE CLASS 10 MATHS NOTES CHAPTER 15 PROBABILITY
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Solution of Linear Equation in two variables:
A pair of numbers, one for x and the other for y, that make the two sides of the equation equal is the solution of a linear equation in two variables.
Solution in Graphics
Graphically representing a pair of linear equations in two variables:
A pair of straight lines can be used to depict a pair of linear equations in two variables graphically.
The following is a graphical method for determining the solution to a pair of linear equations:
Algebraic Solution
Finding a solution to a pair of Linear Equations that are consistent.
The solution of a pair of linear equations is in the form (x,y), which simultaneously solves both equations. It is possible to get a solution for a consistent pair of linear equations by using
Finding a solution to a pair of linear equations using the substitution method:
Method of substitution:
y – 4x = 1
x + 4y= 38
x + 4 (4x + 1) = 38
x + 16x + 4 = 38
17x + 4 = 38
17x = 38-4
17x= 34
X= 2
17 multiplied by 2 equals 34
To find the value of a variable, solve the linear equation in that variable.
X = 2
Y = 4x + 1
Y = 8 + 1
Y = 9
As a result, the solution to the set of linear equations y – 4x = 1 and x + 4y = 38 is (2,9).
Finding a solution to a pair of linear equations using the elimination approach
Method of elimination
Consider the expressions x + 4y = 10 and 2x – y = 2.
Step 1: By multiplying the coefficients of any variable, you can make them the same.
By multiplying the equations with constants, you can achieve the same result. When we divide the first equation by two, we get
20 = 2x + 8y
Step 2: To eliminate one variable, add or subtract the equations, resulting in a single variable equation.
Subtract the second equation from the first.
20 = 2x + 8y
2x – y = 2 – + – ——————– 0(x) + 9y = 18
Step 3: Solve for one variable and use the result to solve for the other variable in any equation.
Y = 2
X = 10 – 4y
X= 10 – 4*2= 10 – 8
X = 2
The answer is (2, 2).
Cross-multiplication A method for solving a pair of linear equations.
In the case of the pair of linear equations
a_{1}x + b_{1}y + c_{1}=0
a_{2}x + b_{2}y + c_{2}=0,
x and y can be calculated as
x = (b_{1}c_{2}−b_{2}c_{1})/(a_{1}b_{2}−a_{2}b_{1})
y = (c_{1}a_{2}−c_{2}a_{1})/(a_{1}b_{2}−a_{2}b_{1})
Using Linear Equations to Solve Problems:
Equations that can be reduced to a pair
Questions in two variables that can be reduced to a pair of Linear Equations.
Some equations can be simplified to a linear equation by substituting one variable for another.
2/x+3/y=4
5/x−4/y=9
In this situation, we can make the change.
1/x is equal to u, and 1/y is equal to v.
2u + 3v = 4 5u – 4v = 9 is the result of the pair of equations.
It is possible to solve the above pair of equations. Back replace the values of x and y after you’ve solved the problem.