sketch the graph for the following linear functions : 1. y=8x+3 / 2. y+1-6x-1 / 3. 3y+2=6x-1/ 4.4y=7x+1

Let's go through each of the linear equations, simplify them (if necessary), and identify their slope-intercept forms so we can sketch their graphs.

### 1. \( y = 8x + 3 \)

This is already in slope-intercept form \( y = mx + b \), where:
- **Slope** \( m = 8 \)
- **Y-intercept** \( b = 3 \)

### 2. \( y + 1 = 6x + 1 \)

We need to simplify this equation:
\[
y = 6x + 1 - 1
\]
\[
y = 6x
\]

So, the slope-intercept form is \( y = 6x \), where:
- **Slope** \( m = 6 \)
- **Y-intercept** \( b = 0 \) (passes through the origin)

### 3. \( 3y + 2 = 6x - 1 \)

Simplifying this equation:
\[
3y = 6x - 1 - 2
\]
\[
3y = 6x - 3
\]
Now divide by 3:
\[
y = 2x - 1
\]

So, the slope-intercept form is \( y = 2x - 1 \), where:
- **Slope** \( m = 2 \)
- **Y-intercept** \( b = -1 \)

### 4. \( 4y = 7x + 1 \)

Dividing both sides by 4:
\[
y = \frac{7}{4}x + \frac{1}{4}
\]

So, the slope-intercept form is \( y = \frac{7}{4}x + \frac{1}{4} \), where:
- **Slope** \( m = \frac{7}{4} \)
- **Y-intercept** \( b = \frac{1}{4} \)

---

To summarize:
1. \( y = 8x + 3 \) (Slope: 8, Y-intercept: 3)
2. \( y = 6x \) (Slope: 6, Y-intercept: 0)
3. \( y = 2x - 1 \) (Slope: 2, Y-intercept: -1)
4. \( y = \frac{7}{4}x + \frac{1}{4} \) (Slope: \( \frac{7}{4} \), Y-intercept: \( \frac{1}{4} \))

Would you like me to generate the graph for these equations?

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Here are 5 follow-up questions:
1. What does the slope tell you about the graph of a linear function?
2. How does changing the y-intercept affect the position of the graph?
3. How can you determine if two lines are parallel based on their equations?
4. What is the significance of a slope of zero in a linear equation?
5. How would you convert a linear equation from standard form to slope-intercept form?

**Tip**: A positive slope means the line rises from left to right, while a negative slope means it falls.

Sketch the graph for the following linear functions: 1. y = 8x + 3 / 2. y + 1 = 6x + 1 / 3. 3y + 2 = 6x - 1 / 4. 4y = 7x + 1

Learn how to graph the linear functions y = 8x + 3, y = 6x, y = 2x - 1, and y = (7/4)x + 1/4. This problem covers key algebraic concepts like slope, y-intercept, and graphing linear equations. Suitable for students in Grades 8-10.

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