https://file.aiquickdraw.com/mathbot/file/1726152798969wetjy1pf.jpg

Find a linear equation whose graph is the straight line with the given properties.

We are tasked with finding the equation of a line that passes through the point \( \left( \frac{1}{7}, -1 \right) \) and is parallel to the line given by the equation:

\[
7x - 8y = 7
\]

### Step 1: Find the slope of the given line
To find the slope of the line \( 7x - 8y = 7 \), we first rewrite it in slope-intercept form, \( y = mx + b \), where \( m \) is the slope.

\[
7x - 8y = 7
\]

Solve for \( y \):

\[
-8y = -7x + 7
\]

\[
y = \frac{7}{8}x - \frac{7}{8}
\]

Thus, the slope of the given line is \( m = \frac{7}{8} \).

### Step 2: Equation of the line parallel to the given line
Since the lines are parallel, the slope of the line we are finding is also \( \frac{7}{8} \).

Using the point-slope form of the equation of a line:

\[
y - y_1 = m(x - x_1)
\]

where \( m = \frac{7}{8} \) and \( (x_1, y_1) = \left( \frac{1}{7}, -1 \right) \), we plug in the values:

\[
y - (-1) = \frac{7}{8} \left( x - \frac{1}{7} \right)
\]

Simplifying:

\[
y + 1 = \frac{7}{8} \left( x - \frac{1}{7} \right)
\]

To simplify further, distribute the \( \frac{7}{8} \):

\[
y + 1 = \frac{7}{8}x - \frac{7}{56}
\]

Since \( \frac{7}{56} = \frac{1}{8} \), the equation becomes:

\[
y + 1 = \frac{7}{8}x - \frac{1}{8}
\]

Finally, subtract 1 from both sides to isolate \( y \):

\[
y = \frac{7}{8}x - \frac{1}{8} - 1
\]

\[
y = \frac{7}{8}x - \frac{1}{8} - \frac{8}{8}
\]

\[
y = \frac{7}{8}x - \frac{9}{8}
\]

### Final Answer:
The equation of the line is:

\[
y = \frac{7}{8}x - \frac{9}{8}
\]

Would you like more details on any part of the solution?

---

Here are some follow-up questions:

1. How can we verify if two lines are parallel based on their slopes?
2. What is the general formula for converting a line equation to slope-intercept form?
3. Can you derive the point-slope form of a line equation from the slope-intercept form?
4. What is the significance of the point \( \left( \frac{1}{7}, -1 \right) \) in this problem?
5. How would the problem change if the lines were perpendicular instead of parallel?

**Tip**: Always remember that parallel lines share the same slope, but have different intercepts!

Find a linear equation whose graph is the straight line with the given properties: through (1/7, -1) and parallel to the line 7x - 8y = 7.

Learn how to find a linear equation that passes through the point (1/7, -1) and is parallel to the line 7x - 8y = 7. This algebra problem covers linear equations, slope-intercept form, and the concept of parallel lines. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation