Rendu chara rashulalo rekhiya sameekaranala Jatha: Notes

Chapter summary: in english

1)Two linear equations within the same two variables are called a pair of linear equations in two variables
a₁x+b₁y+c₁=0(a₁²+b₁²≠0)
a₂x+b₂y+c₂=0(a₂²+b₂²≠0)
where a₁,a₂,b₁,b₂,c₁,c₂ are real numbers.

2) A pair of linear equations in two variables may be solved by using many methods.

3) The graph of a pair of linear equations in two variables is represented by 2 lines.
i. If the lines intersect at a point then the point gives the unique solution of the two equations. In this case, the pair of equations is consistent and independent.
ii. If the lines coincide, then there are infinitely many solutions -where each point on the line could be a solution. In this case, the pair of equations is consistent and dependent.
iii. . If the lines coincide, then there are infinitely many solutions -where each point on the line is a solution. In this case, the pair of equations is consistent and dependent.

4)We have discussed the subsequent methods for locating the solution(s) of a pair of linear equations
i. Model method
ii.Graph method
iii.Algebraic methods - Substitution method and the Elimination method.

5)There is a relation between the coefficients and nature of system of equations.
i. if (a₁ / a₂) ≠ (b₁ / b₂) then the pair of linear equations is consistent.
ii. if (a₁ / a₂) = (b₁ / b₂) ≠ (c₁ / c₂) then the pair of linear equations is inconsistent.
iii. if (a₁ / a₂) = (b₁ / b₂) = (c₁ / c₂) then the pair of linear equations is dependent and consistent.

6) There are several conditions which are also mathematically represented by two equations that don't seem to be linear to start out. But we are able to alter them so that they are goinng to be reduced to a pair of linear equations.

Try these questions

1) For what value of k, the subsequent system of equations has a unique solution. x – ky = 2 and 3x + 2y = –5

2) For what values of m, the pair of equations 3x + my = 10 and 9x + 12y = 30 gives a unique solution.

3) In a rectangle ABCD, AB = x + y, BC = x – y, CD = 9 and AD = 3. Find the values of x and y.

4) Show that the pair Linear Equations 7x + y = 10 and x + 7y = 10 are consultant.

5) Write the Condition for the pair of linear equations in two variables to be parallel lines.

6) If we multiply or divide either sides of a linear equation by a non- zero number, then the roots of that linear equation will remain the same’. is it's true? ? If it is true,then justify with an example.

7) If the current ages of A and B are in ratio of 9 : 4 and after 7 years the ratio of the ages are going to be 5 : 3 then find their current ages.

8) Solve the subsequent pair of linear equations by substitution method. 2x – 3y = 19 and 3x – 2y = 21