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The image presents two perpendicular lines, and the task is to find the slope-intercept form of the equation of line \( y_2 \), based on the equation given for line \( y_1 \). The equation of \( y_1 \) is:

\[
y_1 = 3x - 4
\]

### Key Information:
- The lines are perpendicular, meaning the product of their slopes must equal \( -1 \).
- The slope of \( y_1 \) is \( 3 \) (since the equation is in the form \( y = mx + b \), where \( m \) is the slope).
- For perpendicular lines, if the slope of one line is \( m_1 \), the slope of the other line is \( m_2 = -\frac{1}{m_1} \).

Thus, for line \( y_2 \), the slope \( m_2 \) is:

\[
m_2 = -\frac{1}{3}
\]

The equation of line \( y_2 \) is also in the form \( y = mx + b \). Now, to find the y-intercept \( b_2 \), we can use the point where the two lines intersect, which is given as \( (3, 5) \). Using this point in the equation \( y_2 = -\frac{1}{3}x + b_2 \):

\[
5 = -\frac{1}{3}(3) + b_2
\]
\[
5 = -1 + b_2
\]
\[
b_2 = 6
\]

Thus, the equation of \( y_2 \) is:

\[
y_2 = -\frac{1}{3}x + 6
\]

Let me know if you would like further clarification or additional details.

### Related Questions:
1. How do you find the equation of a line if the slope and a point on the line are given?
2. What is the significance of perpendicular lines in terms of their slopes?
3. How can you verify the intersection point of two lines algebraically?
4. What method would you use to find the distance between two lines?
5. How do parallel and perpendicular slopes differ in geometry?

### Tip:
Always remember, for two lines to be perpendicular, their slopes must multiply to \( -1 \). This relationship helps in quickly identifying the slope of the perpendicular line.

The lines shown below are perpendicular. Find the slope-intercept form of the equation of line y2.

Learn how to find the equation of a perpendicular line given the slope of one line. This problem involves calculating the slope-intercept form and applying the perpendicular slopes theorem using the provided intersection point. Suitable for students in Grades 8-10.

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