Question

Solve the following systems of equations using elimination methods.
 $ 0.5x + 0.8y = 0.44 $ , $ 0.8x - 0.6y = 0.5 $

Answer 1

Hint: Here, to solve the given system of equations using elimination method, we need to make any one variable of both the equations the same by multiplying or dividing both the given equations with any number. Here, we are going to multiply the first equation with 0.8 and the second equation with 0.5. Then we will subtract the first equation from the second equation and hence, one variable will be eliminated.

Complete step-by-step answer:
In this question, we are given a system of equations which has two linear equations and we need to solve this system of equations and find the value of the variables.
The given system of equations is:
 $ 0.5x + 0.8y = 0.44 $ - - - - - - - - - - - (1)
 $ 0.8x - 0.6y = 0.5 $ - - - - - - - - - - - - (2)
Now, there are multiple ways by which we can solve this system of equations. But we are going to use the elimination method in this question.
Elimination method means we need to eliminate any one of the variables from the given two equations by multiplying or dividing the equations with any constant and then substitute the found variable in any one of two given equations and find the value of the second variable.
Here, if we multiply equation (2) with 0.5 and equation (1) with 0.8, then the variable x can be eliminated. Therefore, multiplying equation (2) with 0.5 and equation (1) with 0.8, we get
 $ \Rightarrow \left( {0.8x - 0.6y} \right) \times 0.5 = 0.5 \times 0.5 $
 $ \Rightarrow 0.4x - 0.3y = 0.25 $ - - - - - - - - (3)
And
 $ \Rightarrow \left( {0.5x + 0.8y} \right) \times 0.8 = 0.44 \times 0.8 $
 $ \Rightarrow 0.4x + 0.64y = 0.352 $ - - - - - - (4)
Now, subtracting equation (3) from equation (4), we get
 $
  \underline
  0.4x + 0.64y = 0.352 \\
   - 0.4x + 0.3y = - 0.25 \\
    \\
  0 + 0.94y = 0.102 \\
  $
 $
   \Rightarrow y = \dfrac{{0.102}}{{0.94}} \\
   \Rightarrow y = 0.108 \;
  $
Now, substitute $ y = 0.108 $ in equation (3), we get
 $
   \Rightarrow 0.4x - 0.3y = 0.25 \\
   \Rightarrow 0.4x - 0.3\left( {0.108} \right) = 0.25 \\
   \Rightarrow 0.4x = 0.25 + 0.0324 \\
   \Rightarrow 0.4x = 0.2824 \\
   \Rightarrow x = \dfrac{{0.2824}}{{0.4}} \\
   \Rightarrow x = 0.706 \;
  $
Hence, $ x = 0.706 $ and $ y = 0.108 $ .
So, the correct answer is “ $ x = 0.706 $ and $ y = 0.108 $ ”.

Note: We can also solve this system of equations using substitution method.
 $ 0.5x + 0.8y = 0.44 $ - - - - (1)
 $ 0.8x - 0.6y = 0.5 $ - - - - - (2)
 $
   \Rightarrow 0.5x + 0.8y = 0.44 \\
   \Rightarrow 0.5x = 0.44 - 0.8y \\
   \Rightarrow x = 0.88 - 1.6y \;
  $
Substituting $ x = 0.88 - 1.6y $ in equation (2), we get
 $
   \Rightarrow 0.8\left( {0.88 - 1.6y} \right) - 0.6y = 0.5 \\
   \Rightarrow 0.704 - 1.28y - 0.6y = 0.5 \\
   \Rightarrow 1.88y = 0.204 \\
   \Rightarrow y = \dfrac{{0.204}}{{1.88}} \\
   \Rightarrow y = 0.108 \;
  $
Now, substitute $ y = 0.108 $ in $ x = 0.88 - 1.6y $ , we get
 $
   \Rightarrow x = 0.88 - 1.6\left( {0.108} \right) \\
   \Rightarrow x = 0.88 - 0.1728 \\
   \Rightarrow x = 0.706 \;
  $