https://file.aiquickdraw.com/mathbot/file/1727661760657bmrwvr33.jpg

Let's solve the given system of equations using the substitution method.

The system of equations is:

\[
\begin{aligned}
3x + 12y &= -5 \quad \text{(1)} \\
x &= -4y - 4 \quad \text{(2)}
\end{aligned}
\]

### Step 1: Substitute Equation (2) into Equation (1)

Since \(x = -4y - 4\) from Equation (2), substitute this expression for \(x\) into Equation (1):

\[
3(-4y - 4) + 12y = -5
\]

### Step 2: Simplify the equation

Distribute the 3 on the left-hand side:

\[
-12y - 12 + 12y = -5
\]

### Step 3: Combine like terms

Notice that the \( -12y \) and \( +12y \) cancel each other out, leaving:

\[
-12 = -5
\]

### Step 4: Analyze the result

The equation \(-12 = -5\) is false, which means there is no solution for this system of equations.

Thus, the system has **no solution**, and the answer is:

\[
\boxed{\text{DNE}, \text{DNE}}
\]

Would you like more details or have any questions? Here are five related questions that can expand this topic:

1. How does the substitution method work in general?
2. How do you know when a system of equations has no solution?
3. What is the graphical interpretation of a system with no solution?
4. Can a system of linear equations have an infinite number of solutions? If so, when?
5. What is the difference between the substitution and elimination methods?

**Tip:** Always simplify your equations fully after substituting to avoid missing special cases like no solutions or infinite solutions.

Solve the following system of equations with the substitution method:

3x + 12y = -5
x = -4y - 4

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.

Learn how to solve a system of linear equations using the substitution method. This problem involves algebraic concepts and shows why the system has no solution, as the result leads to an inconsistent statement.

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