Facts that Matter: Pair of Linear Equations in Two Variables

# Facts that Matter: Pair of Linear Equations in Two Variables | Mathematics (Maths) Class 10 PDF Download

### Linear Equation in Two Variable

An equation that can be put in the form
ax + by + c = 0
where a, b and c are real numbers (a, b ≠ 0) is called a linear equation in two variables ‘x’ and ‘y’.

### General Form of a Pair of Linear Equations in Two Variable

The general form of a linear pair of equations in two variables is:
a1x + b1y + c= 0  and
a2x + b2y + c2 = 0
where a1, b1, c1, a2, b2, c2 are real numbers such that the coefficients of x and y cannot be zero simultaneously.

## Solution of a Pair of Linear Equations in Two Variables

The solution of a linear equation in two variables ‘x’ and ‘y’ is a pair of values (one for ‘x’ and other for ‘y’) which makes the two sides of the equation equal.
There are two methods to solve a pair of linear equations:
(i) Algebraic Method
(ii)Graphical Method.

### Algebraic Method

There are 3 algebraic methods for solving a pair of linear equations in two variables.

Types of Algebraic Methods

(i) Substitution Method

Step 1: Find the value of one variable, say y in terms of the other variable, i.e., x from either equation, whichever is convenient

Step 2: Substitute this value of y in the other equation, and reduce it to an equation in one variable, i.e., in terms of x, which can be solved. Sometimes, you will get statements with no variables. If this statement is true, you can conclude that the pair of equations has infinitely many solutions. If the statement is false, then the pair of linear equations is inconsistent.

Step 3: Substitute the value of x (or y) obtained in Step 2 in the equation used in Step 1 to obtain the value of the other variable.

(ii) Elimination Method

Step 1: First, multiply both the equations by some suitable non-zero constants to make the coefficients of one variable (either x or y) numerically equal.

Step 2: Then add or subtract one equation from the other so that one variable gets eliminated. After you get an equation in one variable, go to step 3.

If in Step 2, you obtain a true statement involving no variable, then the original pair of equations has infinitely many solutions.

If in Step 2, you obtain a false statement involving no variable, then the original pair of equations has no solution, i.e., it is inconsistent.

Step 3: Solve the equation in one variable(x or y) so obtained to get its value.

Step 4: Substitute this value of x ( or y) in either of the original equations to get the value of the other variable.

### Graphical Method

(i) If the graphs of two equations of a system intersect at a point, the system is said to have a unique solution, i.e., the system is consistent.
(ii) If the graphs of two equations of a system are two parallel lines, the system is said to have no solution, i.e., the system is inconsistent.
(iii) When the graphs of two equations of a system are two coincident lines, the system is said to have infinitely many solutions, i.e., the system is consistent and dependent.

### Consistent and Inconsistent System

If
form a pair of linear equations, then the following three situations can arise:
(i)   If, then the system is consistent.
(ii)  If , then the system is inconsistent.
(iii)  If  , then the system is dependent and consistent.

The document Facts that Matter: Pair of Linear Equations in Two Variables | Mathematics (Maths) Class 10 is a part of the Class 10 Course Mathematics (Maths) Class 10.
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## FAQs on Facts that Matter: Pair of Linear Equations in Two Variables - Mathematics (Maths) Class 10

 1. What is a pair of linear equations in two variables?
Ans. A pair of linear equations in two variables is a system of two equations that involve two variables, typically represented as x and y, and have the form ax + by = c, where a, b, and c are constants.
 2. How can we solve a pair of linear equations in two variables?
Ans. There are multiple methods to solve a pair of linear equations in two variables, such as substitution method, elimination method, and graphical method. These methods involve manipulating the equations to eliminate one variable and find the values of the remaining variable.
 3. What is the substitution method for solving a pair of linear equations in two variables?
Ans. In the substitution method, one equation is solved for one variable in terms of the other variable. Then, the obtained expression is substituted into the other equation. This process allows us to solve for one variable and substitute its value back into the original equations to find the values of both variables.
 4. How does the elimination method work for solving a pair of linear equations in two variables?
Ans. The elimination method involves manipulating the equations in such a way that adding or subtracting them results in the elimination of one variable. By adding or subtracting the equations, the coefficients of one variable are chosen in such a manner that they cancel out when combined. This leaves a single equation with only one variable, which can be easily solved.
 5. What is the graphical method for solving a pair of linear equations in two variables?
Ans. The graphical method involves representing the two equations on a coordinate plane as lines. The intersection point of these lines represents the solution to the pair of linear equations. By plotting the lines and finding their point of intersection, we can determine the values of the variables.

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