Question

How do you solve $ 5x + 6y = 24 $ and $ 3x + 5y = 18 $ ?

Answer 1

Hint: The given equations is a pair of linear equations in two variables $ x $ and $ y $ . Solving both the equations simultaneously means finding such values of $ x $ and $ y $ for which equality in both the given equations is satisfied. We will use the substitution method to find the solution, i.e., simplifying one variable in the form of another from one equation and then substituting it in the second equation.

Complete step-by-step answer:
The two given equations are $ 5x + 6y = 24 $ and $ 3x + 5y = 18 $ .
From the first equation we can write $ x $ in terms of all other terms, i.e.
 $
  5x + 6y = 24 \\
   \Rightarrow 5x = 24 - 6y \\
   \Rightarrow x = (\dfrac{{24 - 6y}}{5}) \;
  $
Here we get $ x $ in terms of $ y $ .
Since we have to find a solution to a pair of linear equations, this value of $ x $ should also satisfy the second equation, i.e. we can substitute this value of $ x $ in the second equation upholding the equality of the second equation.
Substituting $ x = (\dfrac{{24 - 6y}}{5}) $ in the second equation $ 3x + 5y = 18 $ , we get,
 $
  3x + 5y = 18 \\
   \Rightarrow 3(\dfrac{{24 - 6y}}{5}) + 5y = 18 \;
  $
Now we have a linear equation in one variable $ y $ which we can solve to find the value of $ y $ .
 $
  3(\dfrac{{24 - 6y}}{5}) + 5y = 18 \\
   \Rightarrow (\dfrac{{3 \times 24 - 3 \times 6y}}{5}) + 5y = 18 \\
   \Rightarrow (\dfrac{{72 - 18y}}{5}) + 5y = 18 \\
   \Rightarrow \dfrac{{72}}{5} - \dfrac{{18y}}{5} + 5y = 18 \;
  $
On rearranging we get,
 $
   \Rightarrow 5y - \dfrac{{18y}}{5} = 18 - \dfrac{{72}}{5} \\
   \Rightarrow y(5 - \dfrac{{18}}{5}) = \dfrac{{18 \times 5 - 72}}{5} \\
   \Rightarrow y(\dfrac{{5 \times 5 - 18}}{5}) = \dfrac{{90 - 72}}{5} \\
   \Rightarrow y(\dfrac{{25 - 18}}{5}) = \dfrac{{18}}{5} \\
   \Rightarrow \dfrac{7}{5}y = \dfrac{{18}}{5} \\
   \Rightarrow y = \dfrac{{18}}{7} \;
  $
Thus we get the value of $ y $ as $ \dfrac{{18}}{7} $ .
Now to find the value of $ x $ we can use the value of $ y $ and put the value in either of the two given equations or we can use the simplified equation of $ x $ in terms of $ y $ , i.e. $ x = (\dfrac{{24 - 6y}}{5}) $ .
Putting $ y = \dfrac{{18}}{7} $ in $ x = (\dfrac{{24 - 6y}}{5}) $ , we get:
\[
  x = (\dfrac{{24 - 6 \times \dfrac{{18}}{7}}}{5}) \\
   \Rightarrow x = (\dfrac{{24 - \dfrac{{108}}{7}}}{5}) \\
   \Rightarrow x = (\dfrac{{\dfrac{{24 \times 7 - 108}}{7}}}{5}) \\
   \Rightarrow x = (\dfrac{{\dfrac{{168 - 108}}{7}}}{5}) \\
   \Rightarrow x = \dfrac{{60}}{{7 \times 5}} = \dfrac{{60}}{{35}} = \dfrac{{12}}{7} \;
 \]
Thus, we get the values $ x = \dfrac{{12}}{7} $ and $ y = \dfrac{{18}}{7} $ as the solution for the given pair of linear equations in two variables.
So, the correct answer is “ $ x = \dfrac{{12}}{7} $ and $ y = \dfrac{{18}}{7} $ ”.

Note: Solving a particular equation with two variables is not possible as we have two unknowns with one condition. Solving for two variables requires two independent conditions or equations. We can check the result obtained by putting the values in the two given equations. The result is correct only if the values of $ x $ and $ y $ satisfy both the given equations.