Write an equation of the line containing the given point and perpendicular to the
given line. Express your answer in the form y = mx + b.
(3,9); 3x + y =4

To find the equation of a line that passes through the point \((3, 9)\) and is perpendicular to the given line \(3x + y = 4\), we need to follow these steps:

### Step 1: Find the slope of the given line
First, rewrite the equation of the given line \(3x + y = 4\) in slope-intercept form (\(y = mx + b\)), where \(m\) is the slope.

\[
3x + y = 4
\]
\[
y = -3x + 4
\]

So, the slope of the given line is \(-3\).

### Step 2: Find the slope of the perpendicular line
The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. So, the slope \(m_{\text{perpendicular}}\) of the line perpendicular to \(y = -3x + 4\) is:

\[
m_{\text{perpendicular}} = \frac{1}{3}
\]

### Step 3: Use the point-slope form of the equation
We now know that the slope of the perpendicular line is \(\frac{1}{3}\), and the line passes through the point \((3, 9)\). Using the point-slope form of a line equation:

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

Substitute \(m = \frac{1}{3}\), \(x_1 = 3\), and \(y_1 = 9\):

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

### Step 4: Simplify to get the equation in slope-intercept form
Distribute and simplify the equation:

\[
y - 9 = \frac{1}{3}x - 1
\]
\[
y = \frac{1}{3}x - 1 + 9
\]
\[
y = \frac{1}{3}x + 8
\]

### Final Answer:
The equation of the line perpendicular to \(3x + y = 4\) and passing through the point \((3, 9)\) is:

\[
y = \frac{1}{3}x + 8
\]

Do you need further details or have any questions?

Here are 5 related questions to explore further:
1. How do you find the slope of a line given two points?
2. What is the point-slope form of a line, and when is it useful?
3. How do you convert an equation from standard form to slope-intercept form?
4. What does it mean for two lines to be perpendicular?
5. How can you determine the y-intercept of a line without graphing?

**Tip:** The slope of perpendicular lines are always negative reciprocals. If one slope is \(m\), the other is \(-\frac{1}{m}\).

Write an equation of the line containing the given point and perpendicular to the given line. Express your answer in the form y = mx + b. (3,9); 3x + y = 4

Find the equation of a line that passes through the point (3, 9) and is perpendicular to the line 3x + y = 4. This step-by-step solution uses algebraic methods to determine the slope of perpendicular lines and express the equation in slope-intercept form. Ideal for Grades 9-10 students.

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