https://file.aiquickdraw.com/mathbot/file/1726169782664iguq1z47.jpg

Let's solve this system of equations step-by-step.

The given system of equations is:

\[
y = \frac{1}{3}x + 4 \quad \text{(1)}
\]
\[
3y - x = 12 \quad \text{(2)}
\]

### Substitution Method:
1. From equation (1), we already have an expression for \( y \). Now, we can substitute this value of \( y \) into equation (2).

Substitute \( y = \frac{1}{3}x + 4 \) into equation (2):
\[
3\left(\frac{1}{3}x + 4\right) - x = 12
\]

2. Simplify the left side:
\[
x + 12 - x = 12
\]

3. After simplification:
\[
12 = 12
\]

This equation is true for all values of \( x \), meaning that the system of equations is **dependent**, and the two equations represent the same line. Therefore, there are **infinitely many solutions**.

### Conclusion:
The correct answer is: **There are infinitely many solutions.**

Would you like more details or have any questions? Here are 5 related questions:

1. What does it mean for a system of equations to have infinitely many solutions?
2. How can you tell if a system of equations is dependent or independent?
3. What is the geometric interpretation of dependent equations?
4. Can substitution always be used to solve a system of equations? Why or why not?
5. How would the solution change if the system had no solutions?

**Tip:** When solving systems of equations, always check if they are consistent, inconsistent, or dependent by analyzing their graphical relationships or using substitution/elimination methods.

What is the best way to describe the solution of this system of equations?
y = 1/3x + 4
3y - x = 12

Understand how to solve a system of linear equations using the substitution method. This guide covers identifying dependent systems, interpreting infinitely many solutions, and simplifying equations step-by-step. Suitable 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