AP SSC Class 10 Maths Chapter 4 Pair of Linear Equations in Two Variable

Pair of Linear Equations in Two Variable

Two linear equations that have the same two variables are known as a pair of linear equations in two variables.

\(a_{1}x+b_{1}x+c_{1}=0\,(a_{1}^2+b_{1}^2\neq 0)\) \(a_{2}x+b_{2}x+c_{2}=0\,(a_{2}^2+b_{2}^2\neq 0)\)

where \(a_1,a_2,b_1,b_2,c_1,c_2\) are all real numbers.

We use the following methods to find solutions to a pair of linear equations:

  • Model Method
  • Graphical Method
  • Algebraic methods – Substitution method and Elimination method

There exists a relation between the coefficients and nature of the system of equations. Following are the relationship:

  • \(\frac{a_1}{a_2}\neq \frac{b_1}{b_2}\), then the pair of linear equations is consistent
  • \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq \frac{c_1}{c_2}\), then the pair of linear equations is inconsistent.
  • \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\), then the pair of linear equations is dependent and consistent.

Let us look at a few solved questions from the chapter to better understand a pair of linear equations.

Class 10 Maths Chapter 4 Pair of Linear Equations in Two Variable Solutions

  1. Solve the following pair of equations by reducing them to a pair of linear equations.
  2. \(\frac{5}{x-1}+\frac{1}{y-2}=2\) \(\frac{6}{x-1}+\frac{3}{y-2}=1\)

    Solution:

    Let us consider the following

    \(\frac{1}{x-1}=u\) ……….(1)

    \(\frac{1}{y-2}=v\)………….(2)

    Hence, the equations becomes

    \(5u+v=2\)…………..(3)

    \(6u+3v=1\)…………(4)

    From (1),

    \(v=2-5u\)

    Substituting the above in (4), we get

    \(6u-3(2-5u)=1\)

    Solving,

    \(6u-6+15u=1\) \(21u-6=1\) \(21u=7\) \(u=\frac{7}{21}=\frac{1}{3}\)

    To find the value of v, substitute the above value of u in equation (3)

    \(5(\frac{1}{3})+v=2\) \(\frac{5}{3}+v=2\) \(v=2-\frac{5}{3}\) \(v=\frac{6-5}{3}=\frac{1}{3}\)

    Hence,

    \(v=\frac{1}{3}\)

    To find the value of x and y, substitute the values of u and v in (1) and (2),

    \(u=\frac{1}{x-1}\) \(\frac{1}{3}=\frac{1}{x-1}\) \(x-1=3\) \(x=4\) \(v=\frac{1}{y-2}\) \(\frac{1}{3}=\frac{1}{y-2}\) \({y-2}=3\) \(y=5\)

The value of x and y are 4 and 5 respectively for the given pair of equations.

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