Question

How do you solve the following system $ - 2x - 5y = 17,3x - y = - 61 $ ?

Answer 1

Hint: In order to determine the solution of a given system of equations having two variables, use the method of elimination of term by eliminating the $ y $ term by multiplying by making the mod of coefficient of $ y $ in both the equations equal. Then apply the operation of addition or subtraction according to the equations obtained to eliminate $ y $ term. Solve the result for $ y $ and put the obtained value of $ x $ in any of the equations given to get the value of $ y $ .

Complete step by step solution:
We are given pair of linear equation in two variables $ - 2x - 5y = 17,3x - y = - 61 $
 $ - 2x - 5y = 17 $ ---(1)
 $ 3x - y = - 61 $ ----(2)
In order to solve the system of equations, we have many methods like, substitution, elimination of term, and cross-multiplication.
Here we will be using an elimination method to eliminate the term having $ y $ from both the equations.
And to do so we have to first make the mod of coefficient of $ y $ in both the equation equal to each.
In the first equation the mod of coefficient of $ y $ is 5, so multiplying both side of the equation (2) with the number 5, we get
 $
\Rightarrow 5\left( {3x - y} \right) = 5\left( { - 61} \right) \\
  15x - 5y = - 305 \;
  $
Now Subtracting the above equation with the equation (1) , we get
 $
  15x - 5y - \left( { - 2x - 5y} \right) = - 305 - 17 \\
  15x - 5y + 2x + 5y = - 322 \;
  $
Combining like terms we get
\[17x = - 322\]
Solving the equation for variable $ x $ by dividing both sides of the equation with the coefficient of $ x $ i.e. $ 17 $
\[
\Rightarrow \dfrac{{17x}}{{17}} = \dfrac{{ - 322}}{{17}} \\
  x = \dfrac{{ - 322}}{{17}} \;
 \]
Hence, we have obtained the value of \[x = \dfrac{{ - 322}}{{17}}\].
Now putting this value of $ x $ in the equation (2) to get the value of \[y\]
 $
\Rightarrow 3\left( { - \dfrac{{322}}{{17}}} \right) - y = - 61 \\
  y = 61 - \dfrac{{966}}{{17}} \\
  y = \dfrac{{1037 - 966}}{{17}} \\
  y = \dfrac{{71}}{{17}} \;
  $
Therefore, the solution of system of given equations is $ x = \dfrac{{ - 322}}{{17}},y = \dfrac{{71}}{{17}} $
So, the correct answer is “$ x = \dfrac{{ - 322}}{{17}},y = \dfrac{{71}}{{17}} $”.

Note: Linear Equation in two variable: A linear equation is a equation which can be represented in the form of $ ax + by + c $ where $ x $ and $ y $ are the unknown variables and c is the number known where $ a \ne 0,b \ne 0 $ .
The degree of the variable in the linear equation is of the order 1.