- Books Name
- Mathematics Book for CBSE Class 10
- Publication
- Carrier Point
- Course
- CBSE Class 10
- Subject
- Mathmatics
Basics Revisited
Equation
An equation is a statement that two mathematical expressions having one or more variables are equal.
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 form of a Linear Equation in Two Variables
The general form of a linear equation in two variables is ax + by + c = 0, where a and b cannot be zero simultaneously.
Representing linear equations for a word problem
To represent a word problem as a linear equation
- Identify unknown quantities and denote them by variables.
- Represent the relationships between quantities in a mathematical form, replacing the unknowns with variables.
Solution of a Linear Equation in 2 variables
The solution of a linear equation in two variables is a pair of values, one for x and the other for y, which makes the two sides of the equation equal.
Eg: If 2x+y=4, then (0,4) is one of its solutions as it satisfies the equation. A linear equation in two variables has infinitely many solutions.
Geometrical Representation of a Linear Equation
Geometrically, a linear equation in two variables can be represented as a straight line.
2x – y + 1 = 0
⇒ y = 2x + 1
Graph of y = 2
x
+1
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.
All about Lines
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 straight lines on a plane, there are three possibilities
i) They intersect at exactly one point
pair of linear equations which intersect at a single point.
ii) They are parallel
pair of linear equations which are parallel.
iii) They are coincident
pair of linear equations which are coincident.
Graphical Solution
Representing pair of LE in 2 variables graphically
Graphically, a pair of linear equations in two variables can be represented by a pair of straight lines.
Graphical method of finding solution of a pair of Linear Equations
Graphical Method of finding the solution to a pair of linear equations is as follows:
- Plot both the equations (two straight lines)
- Find the point of intersection of the lines.
The point of intersection is the solution.
Comparing the ratios of coefficients of a Linear Equation
Algebraic Solution
Finding solution for consistent pair of Linear Equations
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
i) Elimination method
ii) Substitution Method
iii) Cross-multiplication method
iv) Graphical method
Substitution Method of finding solution of a pair of Linear Equations
Substitution method:
y – 2x = 1
x + 2y = 12
(i) express one variable in terms of the other using one of the equations. In this case, y = 2x + 1.
(ii) substitute for this variable (y) in the second equation to get a linear equation in one variable, x. x + 2 × (2x + 1) = 12
⇒ 5 x + 2 = 12
(iii) Solve the linear equation in one variable to find the value of that variable.
5 x + 2 = 12
⇒ x = 2
(iv) Substitute this value in one of the equations to get the value of the other variable.
y = 2 × 2 + 1
⇒y = 5
So, (2,5) is the required solution of the pair of linear equations y – 2x = 1 and x + 2y = 12.
Elimination method of finding solution of a pair of Linear Equations
Elimination method
Consider x + 2y = 8 and 2x – 3y = 2
Step 1: Make the coefficients of any variable the same by multiplying the equations with constants. Multiplying the first equation by 2, we get,
2x + 4y = 16
Step 2: Add or subtract the equations to eliminate one variable, giving a single variable equation.
Subtract second equation from the previous equation
2x + 4y = 16
2x – 3y = 2
– + –
———————–
0(x) + 7y =14
Step 3: Solve for one variable and substitute this in any equation to get the other variable.
y = 2,
x = 8 – 2 y
⇒ x = 8 – 4
⇒ x = 4
(4, 2) is the solution.
- Books Name
- Rakhiedu Mathematics Book
- Publication
- Param Publication
- Course
- CBSE Class 10
- Subject
- Mathmatics
3.3 algebraic methods of solving a pair of linear equations
There are four methods for solving a pair of linear equations
(i) Substitution method
(ii) Elimination method
(iii) Cross-multiplication method
(iv) Graphical Method
3.3.2 Elimination Method
Step–I: Obtain the two equations
Step–II: First multiply both the equations by some suitable non-zero constant to make the coefficient of one variable (either x or y) numerically equal.
Step–III: Add or subtract one equation from the other, then one variable gets eliminated.
Step–IV: Solve the equation in one variable.
Step–V: Substitute the value of x (or y) in any one of the given equations and find the value of another variable.
3.3.4 Graphical Method:
In graphical method, we draw the graph of both equations using same pair of horizontal and vertical lines called X-axis and Y-axis respectively. Coordinates of the point(s) of intersection of the two lines is/are the solution.
Nature of solutions:
When we try of solve a pair of equations we could arrive at three possible results. They are, having
(a) a unique solution
(b) an infinite number of solutions
(c) no solution