Question

How do you solve the following system?
$4x + 5y = - 4,$ $ - 3x - 8y = 4$

Answer 1

Hint: According to the given question, we have to solve the given expression of the system as $4x + 5y = - 4,$ $ - 3x - 8y = 4$. So, first of all we have to make the coefficients of either $x$ or $y$ are in the same constant.
So, first of all we have to take L.C.M of the coefficients of either $x$ or $y$ to make the coefficients of either $x$ or $y$ are in the same constant.
Now, we have to add the terms of the given expression of the system which can be any variable or any constant term and same as we have to subtract the terms of the given expression of the system which can be any variable or any constant term.

Complete step-by-step solution:
Step 1: First of all we have to take L.C.M of the coefficients of either $x$ or $y$ to make the coefficients of either $x$ or $y$ are in the same constant.
So, here we have to make the coefficient of $x$ in the same constant by taking the L.C.M of the coefficient $x$.
Step 2: So, we have to multiply by 3 in the given expression $4x + 5y = - 4$ and multiply by 4 in the given expression $ - 3x - 8y = 4$ to make the coefficient of$x$ in the same constant.
$ \Rightarrow 3 \times \left( {4x + 5y = - 4} \right)$ and $4 \times \left( { - 3x - 8y = 4} \right)$
Now, we have to solve the expression obtain just above,
$
   \Rightarrow 3 \times \left( {4x + 5y = - 4} \right) \\
   \Rightarrow 3 \times 4x + 3 \times 5y = 3 \times - 4 \\
   \Rightarrow 12x + 15y = - 12......................(1)
 $
And,
$
   \Rightarrow 4 \times \left( { - 3x - 8y = 4} \right) \\
   \Rightarrow 4 \times - 3x + 4 \times - 8y = 4 \times 4 \\
   \Rightarrow - 12x - 32y = 16......................(2)
 $
Step 3: Now, we have to add the terms of the expression (1) and (2) to obtain the solution step 2 which can be any variable or any constant term and same as we have to subtract the terms of the expression which can be any variable or any constant term.
$
   \Rightarrow 12x - 12x + 15x - 32y = - 12 + 16 \\
   \Rightarrow 0 - 17y = 4 \\
   \Rightarrow - 17y = 4 \\
   \Rightarrow y = - \dfrac{4}{{17}}
 $
Step 4: Now, we have to put the value of $y$ as obtain in the solution step 3 in the expression (1),
$ \Rightarrow 12x + 15\left( {\dfrac{{ - 4}}{{17}}} \right) = - 12$
Now, we have to solve the expression obtain just above
$
   \Rightarrow 12x \times 17 + 15 \times - 4 = - 12 \times 17 \\
   \Rightarrow 204x - 60 = - 204 \\
   \Rightarrow 204x = - 204 + 60 \\
   \Rightarrow 204x = - 144 \\
   \Rightarrow x = - \dfrac{{144}}{{204}} \\
   \Rightarrow x = - \dfrac{{36}}{{51}}
 $

Hence, the solutions of the following system as $4x + 5y = - 4,$ $ - 3x - 8y = 4$ are $x = - \dfrac{{36}}{{51}}$ and $y = - \dfrac{4}{{17}}$

Note: It is necessary to make the coefficients of either $x$ or $y$ are in the same constant.
It is necessary to take L.C.M of the coefficients of either $x$ or $y$ to make the coefficients of either $x$ or $y$ is in the same constant.