Question

How do you solve $ 5x + y = 15, $ $ 2x + y = 12 $ using substitution?

Answer 1

Hint: Here we are given the set of equations in which the left hand side of the equations are not equal and so first make one variable common for both the equations and then we can substitute the values of it in one another equations resulting into one new equations with one variable and then will simplify for the resultant required value.

Complete step by step solution:
Take the given expression: $ 5x + y = 15 $
Move the term with “y” the subject and move the other term on the opposite side. When you move any term from one side to another then the sign of the terms also changes. Positive terms become negative and vice-versa.
 $ y = 15 - 5x $ …. (A)
Now, take other given equation –
 $ 2x + y = 12 $
Make “y” the subject –
 $ y = 12 - 2x $ ….. (B)
Now, take equation (A)
 $ y = 15 - 5x $
Substitute the value of “y” in the above equation from the equation (B)
 $ 12 - 2x = 15 - 5x $
Move term with “x” on the left hand side and constants on the left hand side of the equation –
 $ - 2x + 5x = 15 - 12 $
Simplify the above equation –
 $ 3x = 3 $
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
 $ \Rightarrow x = \dfrac{3}{3} $
Common factors from the numerator and the denominator cancel each other.
 $ \Rightarrow x = 1 $
Place the above value in equation (A)
 $ y = 15 - 5x $
 $ y = 15 - 5(1) $
Simplify the above equation –
 $
  y = 15 - 5 \\
  y = 10 \;
  $
Hence, the required answer is $ (x,y) = (1,10) $
So, the correct answer is “ $ (x,y) = (1,10) $ ”.

Note: Always be careful about the sign convention. When you combine like terms with two different signs, then perform subtraction and give sign of the bigger digit while when you combine with two negative signs then perform addition and give negative sign to the resultant value.