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Solve the following system of equations using the substitution method:

[tex]\[
\begin{cases}
y = -3x - 4 \\
9x + 3y = -12
\end{cases}
\][/tex]

Answer: [tex]\(\square\)[/tex] [tex]\(\square\)[/tex]

Enter your answers as integers or as reduced fractions in the form A/B. If there is no solution, type "DNE" in each blank. If there are an infinite number of solutions, specify their form in the blanks in terms of [tex]\(x\)[/tex].


Sagot :

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]
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