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How many solutions does this system of equations have? Explain how you know.

To determine how many solutions this system of equations has, we first analyze both equations:

1. \( y = -4x + 3 \)
2. \( 2x + 8y = 10 \)

Let's convert the second equation into slope-intercept form (\( y = mx + b \)), where \( m \) is the slope and \( b \) is the y-intercept.

### Step 1: Simplifying the second equation
Start by solving \( 2x + 8y = 10 \) for \( y \):
\[
8y = -2x + 10
\]
\[
y = -\frac{1}{4}x + \frac{5}{4}
\]

Now, we have two equations in slope-intercept form:
1. \( y = -4x + 3 \)
2. \( y = -\frac{1}{4}x + \frac{5}{4} \)

### Step 2: Comparing the slopes
The slopes of the two lines are:
- For the first equation, the slope is \( -4 \).
- For the second equation, the slope is \( -\frac{1}{4} \).

Since the slopes are different, the lines are not parallel and will intersect at exactly one point. Thus, the system of equations has exactly **one solution**.

Would you like more details, or have any questions?

Here are some related questions for further exploration:
1. How would the solution change if the slopes of the two lines were equal?
2. What would happen if the second equation was changed to \( 2x + 8y = 12 \)?
3. Can you determine the exact point of intersection of the two lines?
4. How would the solution be affected if the first equation was modified to \( y = -4x + 5 \)?
5. What does it mean if a system of equations has no solution?

**Tip:** If two linear equations have different slopes, they will always intersect at exactly one point.

Learn how to determine the number of solutions for a system of linear equations with the example y = -4x + 3 and 2x + 8y = 10. This guide explains the slope-intercept method and how to analyze slope differences for unique solutions. Suitable for Grades 8-10 students.

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