Question

Using substitution, solve for x and y in the following equation: $ 2x + y = - 92 $ and $ 2x + 2y = - 98 $ .

Answer 1

Hint: Substitution is a method of solving equations, where we will find the values of variables by putting the value of one variable in place of another. Firstly, we will choose one equation for one of the variables and then we will substitute the value of that variable in another equation to find the value of another variable.

Complete step by step solution:
In this problem, we have given two equations i.e, $ 2x + y = - 92 $ and $ 2x + 2y = - 98 $ and we have to solve the equations to find x and y by the method of substitution, on carefully looking to these equations, we will find the term $ 2x $ is common in both equations, so, we will equate the part of equation to $ 2x $ and then the equations will become,
 $
   \Rightarrow 2x = - 92 - y......................................\left( 1 \right) \\
   \Rightarrow 2x = - 98 - 2y....................................\left( 2 \right) \;
  $
Now, we have the values of $ 2x $ from both the equations and we know that, $ 2x = 2x $ , so, now, we will equate both the equations. On equating the values of $ 2x $ from both the equations, we get,
 $ \Rightarrow - 92 - y = - 98 - 2y..............................\left( 3 \right) $
Now, we will solve this, by adding $ 92 $ on both sides, on adding, we get,
 $ \Rightarrow - 92 - y + 92 = - 98 - 2y + 92 $
On further solving, we get,
 $ \Rightarrow - y = - 6 - 2y $
Now, we will add $ 2y $ on both sides, on adding, we get,
 $ \Rightarrow - y + 2y = - 6 - 2y + 2y $
On further solving, we get,
 $ \Rightarrow y = - 6 $
Now, we have the value of y, so we will substitute it in equation (1) to find the value of x. On substituting we get,
 $
   \Rightarrow 2x = - 92 - y \\
   \Rightarrow 2x = - 92 - \left( { - 6} \right) \\
   \Rightarrow 2x = - 86 \\
  $
On further solving, we get,
 $
   \Rightarrow x = - \dfrac{{86}}{2} \\
   \Rightarrow x = - 43 \;
  $
Hence, the value of y is $ - 6 $ and the value of x is $ - 43 $ .
So, the correct answer is “x = -43 and y =-6”.

Note: Here, we have substituted the value of y in the equation (1) to find the value of x, we can also substitute the value of y in the equation (2), as both the equation provides the same value of x . Let us put the value of y in the equation (2),
 $
   \Rightarrow 2x = - 98 - 2y \\
   \Rightarrow 2x = - 98 - 2 \times \left( { - 6} \right) \\
   \Rightarrow 2x = - 98 + 12 \\
   \Rightarrow 2x = - 86 \\
   \Rightarrow x = \dfrac{{ - 86}}{2} \\
   \Rightarrow x = - 43 \;
  $
Hence, it results the same.