Linear equations involving two unknown variabels are known as two variable equations. Two variables linear equations can be solved either by elimination or substitution method. Let the set of equations is : \[3x + 4y = 17 \]\[2x – 4y = 3\] To solve these type of equations we need to eliminate one unknown form the equations. The elimination can be done by addition, subtraction or by taking any linear combination of the given equations. 

Adding the above two equations, we get

\(5x  =  20 \Rightarrow\  x  = 4\) 

Now putting the value of \(x\) in the first equation (or second equation), we get value of \(y\) 

\(12x + 4y = 17\)

or  \(4y = 5 \Rightarrow \) \(y = \frac{5}{4}\) 

Example: Find the values of \(x\) and \(y\) if \(3x + 4y = 11\) and  \(4x – 2y = 7\)

Solution: The given equations can be written as 

\(3x + 4y - 11 = 0\) and 
\(4x - 2y - 7  = 0\)

Multiplying second equation by 2 and adding to the first equation, we have

\(11x -25=0\) or \(x=\frac{25}{11}\).

Putting the value of \(x\) in the second equation, we have \(y=\frac{23}{22}\)

Three Variables Linear Equations:

Suppose the set of equations is 

\(x + y + z = 6\)  …(1)

\(2x + 3 y - 3z = -1\) …(2)

\(3x + 9y - z    = 18\)  ....(3)

Now from equations (1) & (2) we can eliminate \(z\),multiplying equation (1) by 3 and adding equation (1) & equation (2), we get \(5x+6y=17\)       .....(4)

By adding equation (1) & equation (3), we get 

\(4x + 10y = 24\)  or 

\(2x + 5y = 12\)  …(5)

Now we get only 2 equations (4) & (5) with 2 variables. 

Now   equation \((4) \times 2 - \) equation \((5) \times 5\)

\(\begin{array}{l}(5x + 6y = 17) \times 2\\\underline {(2x + 5y = 12) \times 5\,\,\,\,\,\,\,\,\,\,\,\,} \\- 13y = 34 - 60 =  - 26\end{array}\) or   \(y = 2\) 

Putting value of y in equation (4), we get, 

\(5x + 6 \times 2 = 17\)or \(x = 1\)

Hence   \(x = 1, y = 2\) and  \(z = 3\).

Test on Multivariable linear Equations

 Linear Equations: Lecture 01