## 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

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

**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),

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

Stay tuned to BYJU’S to get the latest notification on SSC exam along with AP SSC model papers, exam pattern, marking scheme and more.