4x+8y=4y−6
Step 2 of 2 :  Find the equation of the line which passes through the point (14,3)
 and is perpendicular to the given line. Express your answer in slope-intercept form. Simplify your answer.

To find the equation of the line that is perpendicular to the given line and passes through the point (14, 3), follow these steps:

### Step 1: Find the slope of the given line.
The given line is:

\[ 4x + 8y = 4y - 6 \]

Start by simplifying this equation. Move all terms involving \( y \) to one side:

\[
4x + 8y - 4y = -6
\]

This simplifies to:

\[
4x + 4y = -6
\]

Now, solve for \( y \) in terms of \( x \) by getting the equation into slope-intercept form \( y = mx + b \):

1. Subtract \( 4x \) from both sides:

\[
4y = -4x - 6
\]

2. Divide through by 4:

\[
y = -x - \frac{3}{2}
\]

So, the slope \( m_1 \) of the given line is \( -1 \).

### Step 2: Find the slope of the perpendicular line.
The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Since the slope of the given line is \( -1 \), the slope of the perpendicular line \( m_2 \) is:

\[
m_2 = \frac{1}{1} = 1
\]

### Step 3: Write the equation of the perpendicular line.
Now, use the point-slope form of a line equation, which is:

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

where \( m = 1 \) (the slope of the perpendicular line) and \( (x_1, y_1) = (14, 3) \) (the point the line passes through). Substituting the values:

\[
y - 3 = 1(x - 14)
\]

Simplify this equation:

\[
y - 3 = x - 14
\]

Add 3 to both sides to get it in slope-intercept form:

\[
y = x - 11
\]

### Final Answer:
The equation of the line that passes through the point (14, 3) and is perpendicular to the given line is:

\[
y = x - 11
\]

Would you like more details or have any further questions?

Here are 5 related questions for deeper understanding:
1. How do you find the equation of a line parallel to a given line through a specific point?
2. What is the geometric interpretation of perpendicular slopes?
3. How do you convert an equation in standard form to slope-intercept form?
4. How do you calculate the distance between a point and a line?
5. How can you find the equation of a line given two points?

**Tip:** When finding a perpendicular slope, always remember to take the negative reciprocal of the original slope.

Find the equation of the line which passes through the point (14,3) and is perpendicular to the given line 4x + 8y = 4y − 6. Express your answer in slope-intercept form. Simplify your answer.

This problem involves finding the equation of a line that is perpendicular to the given line 4x + 8y = 4y - 6 and passes through the point (14, 3). The solution includes key algebraic concepts like slope-intercept form, calculating slopes, and understanding perpendicular lines. Suitable for students in Grades 9-11.

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