Question

How do you solve the system $ 5x + y = 13 $ and $ 2x + 6y = 22? $

Answer 1

Hint: Take the given expressions, and for simplification make the coefficient of one variable same in both the equations and then simplify for the resultant required value.

Complete step by step solution:
Take the given expression: $ 5x + y = 13 $
Multiply the above expression with the number
 $ 30x + 6y = 78 $ …. (A)
 $ 2x + 6y = 22 $ ….. (B)
Subtract equation (B) from the equation (A), the left hand side of both the equations and in the right hand side of the equations.
 $ (30x + 6y) - (2x + 6y) = 78 - 22 $
When there is a negative sign outside the bracket, then the sign of the terms inside the bracket also changes. Positive terms become negative and vice-versa.
 $ (30x + 6y - 2x - 6y) = 56 $
Combine like terms together.
 $ \underline {30x - 2x} + \underline {6y - 6y} = 56 $
Like terms with the same value and opposite sign cancels each other.
 $ \Rightarrow 28x = 56 $
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
 $ \Rightarrow x = \dfrac{{56}}{{28}} $
Common factors from the numerator and the denominator cancels each other.
 $ \Rightarrow x = 2 $
Place the above value in the equation (B)
 $ 2(2) + 6y = 22 $
Simplify the above equation –
 $ 4 + 6y = 22 $
Make “y” the subject –
 $
  6y = 22 - 4 \\
  6y = 18 \\
  y = \dfrac{{18}}{6} \;
  $
Common factors from the numerator and the denominator cancels each other.
 $ \Rightarrow y = 3 $
Hence, the required solutions are - $ (x,y) = (2,3) $
So, the correct answer is “ $ (x,y) = (2,3) $ ”.

Note: Always remember that when we expand the brackets or open the brackets, the sign outside the bracket is most important. If there is a positive sign outside the bracket then the values inside the bracket does not change and if there is a negative sign outside the bracket then all the terms inside the bracket changes. Positive terms change to the negative and the negative term changes to the positive.