Notes On Algebraic Methods of Solving a Pair of Linear Equations - CBSE Class 10 Maths
             To find the solution to pair of linear equations, graphical method may not always give the most accurate solutions. Especially, when the point representing the solution has non-integral coordinates like (√2 , √3) or (   1 3   -5 4 ).

There are three algebraic methods that can be used to solve a pair of linear equations namely (1) Substitution method (2) Elimination method (3) Cross - multiplication method.

Substitution method:
1. The first step to solve a pair of linear equations by the substitution method is to solve one equation for either of the variables.
2. The choice of equation or variable in a given pair does not affect the solution for the pair of equations.
3. In the next step, we’ll substitute the resultant value of one variable obtained in the other equation and solve for the other variable.
4. In the last step, we can substitute the value obtained of the variable in any one equation to find the value of the second variable.

Elimination method:
1. Multiply the equations with suitable non-zero constants, so that the coefficients of one variable in both equations become equal.
2. Subtract one equation from another, to eliminate the variable with equal coefficients.Solve for the remaining variable.
3. Substitute the obtained value of the variable in one of the equations and solve for the second variable.

Cross - multiplication method:

Let’s consider the general form of a pair of linear equations a1x + b1y + c1 = 0 , and a2x + b2y + c2 = 0.

When a1 divided by a2 is not equal to b1 divided by b2, the pair of linear equations will have a unique solution.

To solve this pair of equations for x and y using cross-multiplication, we’ll arrange the variables x and y and their coefficients a1, a2, b1 and b2, and the constants c1 and c2  as shown below


solution to pair of linear equations, algebraic methods, Cross - multiplication method

⇒ x =  b 1 c 2 - b 2 c 1 a 1 b 2 - a 2 b 1   ⇒ y =  c 1a 2 c 2a 1 a 1 b 2 - a 2 b 1

Solving word problems:
1. Read the problem carefully and identify the unknown quantities. Give these quantities like x, y, r, s, t and so on.
2. Identify the variables to be determined.
3. Read the problem carefully and convert the equations in terms of the variables to be determined.
4.Solve the equations using any one of the above three methods.

We can convert non linear equations in to linear equation by a suitable substitution. Then solve those equations using any one of the above three methods.
             

Summary

             To find the solution to pair of linear equations, graphical method may not always give the most accurate solutions. Especially, when the point representing the solution has non-integral coordinates like (√2 , √3) or (   1 3   -5 4 ).

There are three algebraic methods that can be used to solve a pair of linear equations namely (1) Substitution method (2) Elimination method (3) Cross - multiplication method.

Substitution method:
1. The first step to solve a pair of linear equations by the substitution method is to solve one equation for either of the variables.
2. The choice of equation or variable in a given pair does not affect the solution for the pair of equations.
3. In the next step, we’ll substitute the resultant value of one variable obtained in the other equation and solve for the other variable.
4. In the last step, we can substitute the value obtained of the variable in any one equation to find the value of the second variable.

Elimination method:
1. Multiply the equations with suitable non-zero constants, so that the coefficients of one variable in both equations become equal.
2. Subtract one equation from another, to eliminate the variable with equal coefficients.Solve for the remaining variable.
3. Substitute the obtained value of the variable in one of the equations and solve for the second variable.

Cross - multiplication method:

Let’s consider the general form of a pair of linear equations a1x + b1y + c1 = 0 , and a2x + b2y + c2 = 0.

When a1 divided by a2 is not equal to b1 divided by b2, the pair of linear equations will have a unique solution.

To solve this pair of equations for x and y using cross-multiplication, we’ll arrange the variables x and y and their coefficients a1, a2, b1 and b2, and the constants c1 and c2  as shown below


solution to pair of linear equations, algebraic methods, Cross - multiplication method

⇒ x =  b 1 c 2 - b 2 c 1 a 1 b 2 - a 2 b 1   ⇒ y =  c 1a 2 c 2a 1 a 1 b 2 - a 2 b 1

Solving word problems:
1. Read the problem carefully and identify the unknown quantities. Give these quantities like x, y, r, s, t and so on.
2. Identify the variables to be determined.
3. Read the problem carefully and convert the equations in terms of the variables to be determined.
4.Solve the equations using any one of the above three methods.

We can convert non linear equations in to linear equation by a suitable substitution. Then solve those equations using any one of the above three methods.
             

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