Question

Solve the given equations:
 $ \begin{align}
  & 37x+41y=70 \\
 & 41x+37y=86 \\
\end{align} $

Answer 1

Hint: We will use a substitution method to solve this. First look at the substitution method definition carefully and try to understand it. Try to convert the variable y in terms of x from any equation, by this you get an equation which has only one variable. It is called a single variable equation. Try to keep all variable terms on the left hand side and all constants on the right hand side. Algebraically find the value of a variable using the value as you know relation with other variables just substitute it to get the value of that variable. The pair of values will be your result. Just verify them by substituting into one of the equations.

Complete step-by-step answer:
Substitution method: The method of solving a system equations. It works by solving one of the equations for one of the variables to get in terms of other variables, then plugging this back into another equation, and solving for the other variable. By this you can find both the variables. This method is generally used when there are 2 variables, for more variables it will be tough to solve.
Given equations can be written in the form as given below
 $ \begin{align}
  & 37x+41y=70...................\left( 1 \right) \\
 & 41x+37y=86....................\left( 2 \right) \\
\end{align} $
By subtracting 41y on both sides of equation (1) we get it as
 $ 37x=70-41y $
By dividing with 37 on both sides, we can get it as
 $ x=\dfrac{70-41y}{37}..............\left( 3 \right) $
By substituting this in equation (2), we get it as
 $ 41\left( \dfrac{70-41y}{37} \right)+37y=86 $
By taking least common multiple, we can write it as
 $ \dfrac{41\left( 70-41y \right)+37\left( 37y \right)}{37}=86 $
By cross multiplication, we can write it in the form of
 $ 41\left( 70-41y \right)+1369y=3182 $
By simplifying the bracket, we can write it in the form of
 $ 2870-1681y+1369y=3182 $
By subtracting 2870 on both sides, we can write it as
 $ 1369y-1681y=3182-2870=312 $
By taking y common on left hand side, we can write it as
 $ \left( 1369-1681 \right)y=312 $
By simplifying the left hand side, we can write it as
 $ -312y=312 $
By dividing with -312 on both sides, we can write it as
 $ y=\dfrac{312}{-312}=-1 $
By substituting this in equation (3), we can write it as
 $ x=\dfrac{70-41\left( -1 \right)}{37}=\dfrac{111}{37} $
By simplifying the above equation, we can write it as
 $ x=3 $
By above equations we get values of x, y as below
 $ x=3,y=-1 $
By substituting $ x=3,y=-1 $ in equation (2), we can write the solution as
 $ 41\left( 3 \right)+\left( 37 \right)\left( -1 \right)=86 $
By simplifying the equation, we can write it in the form of
 $ 123-37=86 $
By simplifying, we can write it in the form of
 $ 86=86 $
LHS = RHS. Hence proved.
Therefore $ \left( 3,-1 \right) $ is the solution of the given equation.

Note: Be careful while removing brackets, don’t forget that the constant must also be multiplied. Generally students multiply to variable and forget about constant verification of solutions must be done to prove that our result is correct. Similarly you can first find x in terms of y and then substitute and continue. Anyways you will get the same result because the value of x, y won’t change.