What is an equation of the line that passes through the point left parenthesis, 8, comma, minus, 2, right parenthesis(8,−2) and is perpendicular to the line 4, x, minus, 3, y, equals, 274x−3y=27?

To find the equation of a line that passes through the point \((8, -2)\) and is perpendicular to the line \(4x - 3y = 27\), follow these steps:

### 1. Rewrite the given line in slope-intercept form (\(y = mx + b\)):

The general form of a line is \(Ax + By = C\), and we are given the equation \(4x - 3y = 27\). Solve for \(y\):

\[
-3y = -4x + 27
\]
\[
y = \frac{4}{3}x - 9
\]

This is the equation of the given line, where the slope \(m = \frac{4}{3}\).

### 2. Find the slope of the perpendicular line:

The slopes of perpendicular lines are negative reciprocals. So, if the slope of the given line is \(\frac{4}{3}\), the slope of the perpendicular line will be:

\[
m_{\text{perpendicular}} = -\frac{3}{4}
\]

### 3. Use the point-slope form to find the equation of the new line:

The point-slope form of a line is:

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

We know the slope \(m = -\frac{3}{4}\) and the point \((x_1, y_1) = (8, -2)\). Plug these values into the equation:

\[
y - (-2) = -\frac{3}{4}(x - 8)
\]
\[
y + 2 = -\frac{3}{4}(x - 8)
\]

### 4. Simplify the equation:

Distribute the slope on the right side:

\[
y + 2 = -\frac{3}{4}x + 6
\]

Subtract 2 from both sides:

\[
y = -\frac{3}{4}x + 4
\]

Thus, the equation of the line that passes through \((8, -2)\) and is perpendicular to the line \(4x - 3y = 27\) is:

\[
y = -\frac{3}{4}x + 4
\]

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. What is the general form of a line equation?
2. How do you calculate the negative reciprocal of a slope?
3. Can you derive the slope-intercept form from any line equation?
4. What is the significance of perpendicular lines in geometry?
5. How do you determine if two lines are parallel or perpendicular?

**Tip:** Always verify if the given point satisfies the final line equation after finding it.

What is an equation of the line that passes through the point (8,−2) and is perpendicular to the line 4x−3y=27?

Discover how to find the equation of a line passing through the point (8,−2) and perpendicular to the line 4x−3y=27. This step-by-step solution involves slope-intercept form, point-slope form, and the concept of negative reciprocal slopes. Perfect for students in Grades 9-11.

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