https://file.aiquickdraw.com/mathbot/file/1728622648585z73ma440.jpg

The system of equations shown in the image is:

\[
\begin{aligned}
x - 3y &= -5 \quad \text{(Equation 1)} \\
-12y &= -20 - 4x \quad \text{(Equation 2)}
\end{aligned}
\]

We are asked to solve this system using substitution. Let’s go through the steps:

### Step 1: Solve one equation for one variable
From Equation 1, solve for \(x\):
\[
x = 3y - 5 \quad \text{(rearranging Equation 1)}
\]

### Step 2: Substitute into the other equation
Now, substitute \(x = 3y - 5\) into Equation 2:
\[
-12y = -20 - 4(3y - 5)
\]

### Step 3: Simplify the equation
Expand the right-hand side:
\[
-12y = -20 - 4(3y - 5) = -20 - 12y + 20
\]
\[
-12y = -12y
\]

### Step 4: Analyze the result
The equation simplifies to \(-12y = -12y\), which is a **true statement** for all values of \(y\). This indicates that the system has an **infinite number of solutions**, because the two equations represent the same line.

### Final Answer:
The system has **infinite number of solutions**.

Would you like further details or have any questions?

---

Here are 5 related questions:
1. How do you recognize when a system has infinite solutions?
2. What is the difference between dependent and independent systems of equations?
3. Can substitution always be used to solve systems of equations, or are there exceptions?
4. How can you verify if two equations represent the same line?
5. What is the difference between substitution and elimination methods?

**Tip:** When a system simplifies to a true statement (like \(0 = 0\) or \(x = x\)), it means there are infinite solutions, indicating that the lines overlap.

Solve the system using substitution: x - 3y = -5 and -12y = -20 - 4x.

Learn how to solve the system of equations x - 3y = -5 and -12y = -20 - 4x using the substitution method. This problem explores algebraic techniques and properties of equality, ideal for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation