To solve the following system of equations using the substitution method:
[tex]\[
\left\{\begin{array}{l}
y = -3x - 4 \\
9x + 3y = -12
\end{array}\right.
\][/tex]
Follow these steps:
1. Substitute [tex]\( y \)[/tex] in the second equation:
Given the first equation [tex]\( y = -3x - 4 \)[/tex], we substitute [tex]\( y \)[/tex] into the second equation [tex]\( 9x + 3y = -12 \)[/tex].
[tex]\[
9x + 3(-3x - 4) = -12
\][/tex]
2. Simplify the equation:
Distribute the 3 into the terms inside the parentheses.
[tex]\[
9x + 3(-3x) + 3(-4) = -12 \implies 9x - 9x - 12 = -12
\][/tex]
3. Combine like terms:
This simplifies to:
[tex]\[
9x - 9x - 12 = -12 \implies -12 = -12
\][/tex]
Notice that the variable terms [tex]\( 9x \)[/tex] and [tex]\( -9x \)[/tex] cancel each other out, leaving just [tex]\(-12 = -12\)[/tex], which is a true statement.
4. Interpret the result:
Since the final equation [tex]\(-12 = -12\)[/tex] is always true, it indicates that the system of equations has an infinite number of solutions.
5. Express the solutions in terms of [tex]\( x \)[/tex]:
We can express the solutions in the form of [tex]\( y = -3x - 4 \)[/tex] where [tex]\( x \)[/tex] can be any real number. Thus, the solutions are pairs [tex]\((x, y)\)[/tex] that satisfy this relationship.
The solutions to the system of equations are:
[tex]\[
\boxed{\text{infinite solutions in the form } (x, -3x - 4)}
\][/tex]
