CBSE Class 10 Pair of linear equations Important Formulas and concepts for exams

CBSE Class 10 Pair of linear equations Important Formulas and concepts for exams. There are many more useful educational material which the students can download in pdf format and use them for studies. Study material like concept maps, important and sure shot question banks, quick to learn flash cards, flow charts, mind maps, teacher notes, important formulas, past examinations question bank, important concepts taught by teachers. Students can download these useful educational material free and use them to get better marks in examinations.  Also refer to other worksheets for the same chapter and other subjects too. Use them for better understanding of the subjects.

An equation of the form ax + by + c = 0, where a, b and c are real numbers ( a b   0, 0 ), is called a linear equation in two variables x and y.

The numbers a and b are called the coefficients of the equation ax + by + c = 0 and the number c is called the constant of the equation ax + by + c = 0.

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, c₂ are real numbers, such that a₁ 2 + b₁ 2 ≠ 0, a₂ 2 + b₂ 2 ≠ 0.

CONSISTENT SYSTEM

A system of simultaneous linear equations is said to be consistent, if it has at least one solution.

INCONSISTENT SYSTEM

A system of simultaneous linear equations is said to be inconsistent, if it has no solution.  

METHOD TO SOLVE A PAIR OF LINEAR EQUATION OF TWO VARIABLES

A pair of linear equations in two variables can be represented, and solved, by the:

(i) graphical method (ii) algebraic method 

GRAPHICAL METHOD OF SOLUTION OF A PAIR OF LINEAR EQUATIONS

The graph of a pair of linear equations in two variables is represented by two lines.

1.If the lines intersect at a point, then that point gives the unique solution of the two equations. In this case, the pair of equations is consistent.

2.If the lines coincide, then there are infinitely many solutions — each point on the line being a solution. In this case, the pair of equations is dependent (consistent).

3. If the lines are parallel, then the pair of equations has no solution. In this case, the pair of equations is inconsistent.

Algebraic interpretation of pair of linear equations in two variables

The pair of linear equations represented by these lines  a₁x + b₁y + c1 = 0 and a₂x + b₂y + c₂ = 0

1. If a₁/a₂ ≠b₁/b₂ then the pair of linear equations has exactly one solution.

2. If a₁/a₂ = b₁/b₂ = c₁/c₂ then the pair of linear equations has infinitely many solutions.

3. If a₁/a₂ = b₁/b₂ ≠ c₁/c₂ then the pair of linear equations has no solution.

ALGEBRAIC METHODS OF SOLVING A PAIR OF LINEAR EQUATIONS

Substitution Method

Following are the steps to solve the pair of linear equations by substitution method:
a₁x + b₁y + c₁ = 0 … (i) and
a₂x + b₂y + c₂ = 0 … (ii)

Step 1: We pick either of the equations and write one variable in terms of the other
y = a₁/b₁ x - c₁/b₁ ..... (iii)

Step 2: Substitute the value of x in equation (i) from equation (iii) obtained in step 1.

Step 3: Substituting this value of y in equation (iii) obtained in step 1, we get the values of x and y.

Elimination Method

Following are the steps to solve the pair of linear equations by 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.

♦ If you get an equation in one variable, go to Step 3.

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

♦ If in Step 2, we 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.

Cross - Multiplication Method
Let the pair of linear equations be:
a₁x + b₁y + c₁ = 0 … (1) and
a₂x + b₂y + c₂ = 0 … (2)

The arrows between the two numbers indicate that they are to be multiplied and the second product is to be subtracted from the first.

For solving a pair of linear equations by this method, we will follow the following steps :

Step 1 : Write the given equations in the form (1) and (2).

Step 2 : Taking the help of the diagram above, write Equations as given in (3).

Step 3 : Find x and y, provided a₁b₂ - a₂b₁ = 0

Step 2 above gives you an indication of why this method is called the cross-multiplication method.

Please click the link below to download CBSE Class 10 Pair of linear equations Important Formulas and concepts for exams.