Find an equation of the line that is perpendicular to the line y-2= -2/3(x-6) and passes through the point (1,7).


To find the equation of a line that is perpendicular to the line given by \( y - 2 = -\frac{2}{3}(x - 6) \) and passes through the point \( (1,7) \), follow these steps:

### Step 1: Identify the slope of the given line
The given line equation is in point-slope form:  
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope.

In \( y - 2 = -\frac{2}{3}(x - 6) \), the slope \( m = -\frac{2}{3} \).

### Step 2: Find the perpendicular slope
Two lines are perpendicular if the product of their slopes is \(-1\). Thus, the slope \( m_{\text{perpendicular}} \) of the perpendicular line satisfies:
\[
m_{\text{perpendicular}} \cdot -\frac{2}{3} = -1
\]
Solving for \( m_{\text{perpendicular}} \):
\[
m_{\text{perpendicular}} = \frac{3}{2}
\]

### Step 3: Use the point-slope form with the new slope and point \( (1, 7) \)
Now that we know the perpendicular slope \( m_{\text{perpendicular}} = \frac{3}{2} \) and that the line passes through \( (1, 7) \), we can substitute these into the point-slope form:
\[
y - y_1 = m(x - x_1)
\]
\[
y - 7 = \frac{3}{2}(x - 1)
\]

### Step 4: Simplify to slope-intercept form (optional)
Distribute \( \frac{3}{2} \) on the right side:
\[
y - 7 = \frac{3}{2}x - \frac{3}{2}
\]
Add 7 to both sides to solve for \( y \):
\[
y = \frac{3}{2}x + \frac{11}{2}
\]

### Final Answer
The equation of the line that is perpendicular to \( y - 2 = -\frac{2}{3}(x - 6) \) and passes through the point \( (1, 7) \) is:
\[
y = \frac{3}{2}x + \frac{11}{2}
\]

Would you like a breakdown of each step or have any questions?

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Here are 5 related questions for further practice:

1. Find the equation of a line parallel to \( y - 2 = -\frac{2}{3}(x - 6) \) that passes through \( (1, 7) \).
2. Determine the point where the line \( y = \frac{3}{2}x + \frac{11}{2} \) intersects the x-axis.
3. What is the slope of a line perpendicular to a line with slope \( \frac{3}{5} \)?
4. If a line has an equation \( y = -\frac{1}{4}x + 3 \), what is the slope of any line perpendicular to it?
5. Find the distance between the point \( (1,7) \) and the line \( y - 2 = -\frac{2}{3}(x - 6) \).

**Tip:** To quickly check if two lines are perpendicular, multiply their slopes. If the result is -1, the lines are indeed perpendicular.

Find an equation of the line that is perpendicular to the line y-2= -2/3(x-6) and passes through the point (1,7).

Learn how to find the equation of a line that is perpendicular to y - 2 = -2/3(x - 6) and passes through (1,7). This solution covers essential algebraic and geometric concepts, including finding perpendicular slopes and using point-slope form to derive the equation.

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