Question

How do you solve the system \[6x - 2y = 0\] and $x - y = 12$ by substitution?

Answer 1

Hint: According to given in the question we have to determine or solve the system \[6x - 2y = 0\] and $x - y = 12$ by substitution mentioned. So, first of all we have to use the elimination method to eliminate the terms of the expression which is as explained below:
Elimination method: According to elimination method we just have to eliminate the terms of the expression but before that we have to make the given variables same for both of the expressions given which can be done by multiplying or dividing the terms in the given variable to make them as a same term or variable. Then we have to subtract the terms of the expressions so that we can determine the value of any one of the variables.
Now, we have to multiply with 2 in the given expression $x - y = 12$so that we can make the variable y same for both of the expressions.
Now, we have to subtract the both of the expressions so that we can determine the value of the variable x.
Now, we have to substitute the value of the obtained variable x with the help of the substitution method but before that we have to understand about the substitution method which is as explained below:
Substitution method: According to this method we have to substitute the value of the variable obtained in any one of the expressions as mentioned in the question.
Hence, on substituting the value of the variable in any one of the expressions we can determine another variable.

Complete step-by-step answer:
Step 1: First of all, we have to multiply with 2 in the given expression $x - y = 12$ so that we can make the variable y same for both of the expressions as mentioned in the solution hint. Hence,
$
   \Rightarrow 2(x - y) = 12 \times 2 \\
   \Rightarrow 2x - 2y = 24................(1) \\
 $
Step 2: Now, we have to subtract the both of the expressions so that we can determine the value of the variable x which is as mentioned in the solution hint. Hence,
\[
   \Rightarrow 6x - 2y - (2x - 2y) = 0 - 24 \\
   \Rightarrow 6x - 2y - 2x + 2y = - 24 \\
   \Rightarrow 4x = - 24 \\
 \]
Now, on applying the cross-multiplication e can determine the value of x,
\[
   \Rightarrow x = \dfrac{{ - 24}}{4} \\
   \Rightarrow x = - 6 \\
 \]
Step 3: Now, we have to substitute the value of the obtained variable x as obtained in the solution step 2 with the help of the substitution method which is explained in the solution hint. Hence, we have to substitute the value of x in the expression $x - y = 12$ which is given in the question. Hence,
$
   \Rightarrow - 6 - y = 12 \\
   \Rightarrow - y = 12 + 6 \\
   \Rightarrow - y = 18............(2) \\
 $
Step 4: Now, we have to multiply with negative sign in the both sides of the expression (2) which is as obtained in the solution step 3. Hence,
$ \Rightarrow y = - 18$

Final solution: Hence, we have determined the solution of the expressions which are given in the question are \[6x - 2y = 0\] and $x - y = 12$ with the help of the substitution method $x = - 6$ and $y = - 18$.

Note:
To eliminate the terms of the expression but before that we have to make the given variables same for both of the expressions given which can be done by multiplying or dividing the terms in the given variable to make them as a same term or variable.
To determine the value of another variable we have to substitute the value of the variable obtained in any one of the expressions as mentioned in the question.