find the equation of a line which is perpendicular to the line 7x-2=6 and intersects the line 7x-2y=6 at x=8

To find the equation of the line that is perpendicular to the given line \( 7x - 2 = 6 \) and intersects the line \( 7x - 2y = 6 \) at \( x = 8 \), let's proceed step by step.

### Step 1: Find the slope of the line \( 7x - 2 = 6 \)
First, rearrange the given equation \( 7x - 2 = 6 \) to find its slope.

\[
7x - 2 = 6
\]
\[
7x = 8
\]
\[
x = \frac{8}{7}
\]

This is a vertical line, meaning its slope is undefined.

Since the slope of a vertical line is undefined, the slope of a line perpendicular to it will be **0**. Hence, the line we are looking for must be horizontal.

### Step 2: Equation of a horizontal line
A horizontal line has the form \( y = c \), where \( c \) is a constant. To find \( c \), we need to determine the \( y \)-coordinate of the point where the horizontal line intersects the line \( 7x - 2y = 6 \) at \( x = 8 \).

### Step 3: Find the point of intersection with the line \( 7x - 2y = 6 \)
Substitute \( x = 8 \) into the equation \( 7x - 2y = 6 \):

\[
7(8) - 2y = 6
\]
\[
56 - 2y = 6
\]
\[
-2y = 6 - 56
\]
\[
-2y = -50
\]
\[
y = 25
\]

Thus, the point of intersection is \( (8, 25) \).

### Step 4: Write the equation of the line
Since the line is horizontal and passes through \( (8, 25) \), its equation is:

\[
y = 25
\]

### Final Answer:
The equation of the line is \( \boxed{y = 25} \).

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

### Related Questions:
1. How do you find the slope of a line from its equation?
2. What are the properties of perpendicular lines?
3. How do you find the equation of a line given its slope and a point on the line?
4. What is the difference between vertical and horizontal lines?
5. How do you find the point of intersection between two lines?

### Tip:
For any line perpendicular to a vertical line, the slope will always be zero because perpendicular lines will be horizontal!

Find the equation of a line which is perpendicular to the line 7x-2=6 and intersects the line 7x-2y=6 at x=8.

Learn how to find the equation of a line that is perpendicular to 7x-2=6 and intersects the line 7x-2y=6 at x=8. This problem explores key algebraic concepts such as slopes of perpendicular lines, finding points of intersection, and writing equations of lines. Suitable for students in Grades 9-10.

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