Question

How do you solve $y = - {x^2} - 6x - 3,y = 6$ using substitution method?

Answer 1

Hint: According to the question we have to solve the given expression which is $y = - {x^2} - 6x - 3,y = 6$ as mentioned in the question using the substitution method. So, first of all to determine the required solution using substitution method we have to understand about the substitution method which is as explained below:
Substitution method: According to the substitution method we have to substitute the value of any one of the variables in the other expression to determine the value of the other variable and we can also say that we just have to substitute the value of the given variable to the given expression to determine the other variable.
Now, according to the substation method we just have to substitute the value of variable $y$ which is as mentioned in the question $y = 6$ in the expression as given in the question.
Now, we have to rearrange the terms of the quadratic expression after substituting the value of the variable y.
Now, to solve the quadratic expression obtained we have to obtain the coefficient of x with the factors of the coefficient of ${x^2}$ and the constant term.

Complete step-by-step solution:
Step 1: First of all to determine the required solution using substitution method we have to understand about the substitution method which is as explained in the solution hint.
Step 2: Now, according to the substation method we just have to substitute the value of variable $y$ which is as mentioned in the question $y = 6$ in the expression as given in the question. Hence,
$ \Rightarrow 6 = - {x^2} - 6x - 3$
Step 3: Now, we have to rearrange the terms of the quadratic expression after substituting the value of the variable y.
$
   \Rightarrow {x^2} + 6x + 3 + 6 = 0 \\
   \Rightarrow {x^2} + 6x + 9 = 0
 $
Step 4: Now, to solve the quadratic expression obtained we have to obtain the coefficient of x with the factors of the coefficient of ${x^2}$ and the constant term. Hence,
$
   \Rightarrow {x^2} + (3 + 3)x + 9 = 0 \\
   \Rightarrow {x^2} + 3x + 3x + 9 = 0
 $
Now, we have to take x as a common term from the expression as obtained just above,
$
   \Rightarrow x(x + 3) + 3(x + 3) = 0 \\
   \Rightarrow (x + 3)(x + 3) = 0
 $
Step 5: Now, we just have to solve the expression as obtained in the solution step 4 to determine the required solutions. Hence,
$
   \Rightarrow (x + 3) = 0 \\
   \Rightarrow x = - 3
 $
And,
$
   \Rightarrow (x + 3) = 0 \\
   \Rightarrow x = - 3
 $

Hence, we have determined the required solution of the given expression with the help of the substitution method which are $x = - 3, - 3.$

Note: On solving a quadratic expression only two possible roots/zeroes can be obtained which will satisfy the given quadratic expression means on substituting these roots/zeroes will make the expression 0.
In the substitution method we have to substitute the value of any one of the variables in the other expression to determine the value of the other variable.