What is an equation of the line that passes through the point left parenthesis, 1, comma, minus, 7, right parenthesis(1,−7) and is parallel to the line 6, x, plus, y, equals, 86x+y=8?

To find the equation of a line that passes through the point \((1, -7)\) and is parallel to the line \(6x + y = 8\), follow these steps:

### 1. Identify the slope of the given line:
The general form of a line is \(Ax + By = C\). To find the slope, we convert this to the slope-intercept form \(y = mx + b\), where \(m\) is the slope.

Given the equation \(6x + y = 8\), solve for \(y\):
\[
y = -6x + 8
\]
So, the slope \(m\) of this line is \(-6\).

### 2. Use the point-slope form of the equation:
The line we are looking for is parallel to this line, meaning it has the same slope. Therefore, the slope of our line is also \(-6\).

The point-slope form of a line is:
\[
y - y_1 = m(x - x_1)
\]
where \((x_1, y_1)\) is the point the line passes through, and \(m\) is the slope. Using the point \((1, -7)\) and slope \(-6\), we substitute into the formula:
\[
y - (-7) = -6(x - 1)
\]
Simplify:
\[
y + 7 = -6(x - 1)
\]
\[
y + 7 = -6x + 6
\]
\[
y = -6x + 6 - 7
\]
\[
y = -6x - 1
\]

### 3. Final equation:
The equation of the line is:
\[
y = -6x - 1
\]

Would you like further details or have any questions?

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Here are 5 related questions to extend this topic:
1. How do you find the equation of a line perpendicular to another line?
2. What is the significance of the slope in determining parallelism between two lines?
3. How do you convert from point-slope form to slope-intercept form?
4. Can you derive the equation of a line given two points instead of one point and a slope?
5. How do you determine the y-intercept of a line given a point and the slope?

**Tip**: Always check the slope first when determining if two lines are parallel; if their slopes are equal, they are parallel.

What is an equation of the line that passes through the point (1, −7) and is parallel to the line 6x + y = 8?

Find the equation of a line that passes through the point (1, -7) and is parallel to the line 6x + y = 8. This problem explores key concepts in algebra, including the slope-intercept form and point-slope form, suitable for Grades 8-10 students.

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