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To solve this problem, let's break it down step by step:

We are asked to write the equation of a line that passes through the point \((1, 9)\) and is **perpendicular** to the line given by the equation \(3x + 7y = 1\).

### Step 1: Find the slope of the given line.
The equation of the line is in **standard form**: \(3x + 7y = 1\).

To find the slope, we need to rewrite this equation in **slope-intercept form**: \(y = mx + b\), where \(m\) is the slope.

Start by solving for \(y\):

\[
7y = -3x + 1
\]
\[
y = -\frac{3}{7}x + \frac{1}{7}
\]

So, the slope of the given line is \(m = -\frac{3}{7}\).

### Step 2: Find the slope of the perpendicular line.
Two lines are **perpendicular** if the product of their slopes is \(-1\). Let \(m_1\) be the slope of the given line and \(m_2\) be the slope of the perpendicular line. Then:

\[
m_1 \cdot m_2 = -1
\]

Since \(m_1 = -\frac{3}{7}\), we find \(m_2\) as follows:

\[
-\frac{3}{7} \cdot m_2 = -1
\]
\[
m_2 = \frac{7}{3}
\]

So, the slope of the perpendicular line is \(m_2 = \frac{7}{3}\).

### Step 3: Write the equation in slope-intercept form.
Now that we have the slope of the perpendicular line \(m = \frac{7}{3}\) and the point \((1, 9)\), we can use the **point-slope form** to write the equation of the line:

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

Substitute the values:

\[
y - 9 = \frac{7}{3}(x - 1)
\]

Now, simplify the equation:

\[
y - 9 = \frac{7}{3}x - \frac{7}{3}
\]
\[
y = \frac{7}{3}x - \frac{7}{3} + 9
\]
\[
y = \frac{7}{3}x + \frac{20}{3}
\]

So, the equation in **slope-intercept form** is:

\[
y = \frac{7}{3}x + \frac{20}{3}
\]

### Step 4: Convert to standard form.
To convert the equation into **standard form**, we need to eliminate the fractions. Multiply through by 3:

\[
3y = 7x + 20
\]

Now, rearrange the terms to get all variables on one side:

\[
7x - 3y = -20
\]

This is the equation in **standard form**.

### Final Answer:
- The equation in **slope-intercept form** is: \(y = \frac{7}{3}x + \frac{20}{3}\).
- The equation in **standard form** is: \(7x - 3y = -20\).

Would you like any further clarifications on this? Here are some related questions you can explore:

1. How do you find the slope of a line from its equation in standard form?
2. What is the relationship between the slopes of perpendicular lines?
3. Can a line be both parallel and perpendicular to another line?
4. How do you convert between slope-intercept and standard forms of a line equation?
5. What is the significance of the point-slope form in determining the equation of a line?

**Tip:** When converting an equation from slope-intercept form to standard form, make sure to eliminate fractions by multiplying through by the least common denominator.

Write an equation (a) in slope-intercept form and (b) in standard form for the line passing through (1, 9) and perpendicular to 3x + 7y = 1.

Solve a line equation problem where the line is perpendicular to 3x + 7y = 1 and passes through the point (1,9). Learn how to find the slope-intercept and standard forms of the line equation using coordinate geometry. Suitable for high school students in Grades 9-12.

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