https://file.aiquickdraw.com/mathbot/file/1727146541838pdpaz6h2.jpg

Let's break down the problem from the image you've uploaded. The task is to solve a system of linear equations by graphing.

The system of equations given is:
\[
6x + 3y = 24 \quad \text{(Equation 1)}
\]
\[
8x + 4y = 32 \quad \text{(Equation 2)}
\]

### Step 1: Convert each equation to slope-intercept form (y = mx + b).
1. **For the first equation (6x + 3y = 24):**

   \[
   3y = -6x + 24
   \]
   Divide by 3:
   \[
   y = -2x + 8
   \]

2. **For the second equation (8x + 4y = 32):**

   \[
   4y = -8x + 32
   \]
   Divide by 4:
   \[
   y = -2x + 8
   \]

### Step 2: Analyze the system.
Both equations are actually the same after simplification:
\[
y = -2x + 8
\]
This means the two lines are identical, so they overlap.

### Step 3: Conclusion.
Since the two equations represent the same line, the system has **infinitely many solutions**. The solutions are all the points on the line \( y = -2x + 8 \).

The system is **consistent** and **dependent**, meaning that every point on the line is a solution.

Would you like more details, or do you have any questions? 

Here are some related questions to expand your understanding:
1. How do you determine if two lines are parallel by comparing equations?
2. What does it mean if two lines intersect at one point when graphing a system of equations?
3. How can you solve a system of equations algebraically instead of by graphing?
4. What is the significance of a consistent and dependent system in real-world applications?
5. How would the graph change if one of the slopes was different from the other?

**Tip:** When solving systems by graphing, always convert the equations to slope-intercept form for easier graphing!

Solve the following system of linear equations by graphing: 6x + 3y = 24 and 8x + 4y = 32.

Learn how to solve the system of linear equations 6x + 3y = 24 and 8x + 4y = 32 by converting each equation into slope-intercept form. Discover the solution through graphing and explore consistent and dependent systems in linear algebra. Ideal for students in Grades 8-10.

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