Question

Write an equation of the line passing through the point (2,-4) and perpendicular to the line \(3 x-2 y=6\)

Short answer

Answer: The equation of the line is \(y = -\frac{2}{3}x - \frac{8}{3}\).

Step-by-step solution

Find the slope of the given line

To find the slope of the line \(3x - 2y = 6\), we'll rewrite it in slope-intercept form (\(y = mx + b\)). First, subtract \(3x\) from both sides of the equation, then divide the equation by -2: -2y = -3x + 6 y = \frac{3}{2}x - 3 The slope of the given line is \(\frac{3}{2}\).

Find the slope of the perpendicular line

When two lines are perpendicular, the product of their slopes is -1. Let's call the slope of the perpendicular line \(m_p\): \(m_p \cdot \frac{3}{2} = -1\) To find \(m_p\), divide both sides of the equation by \(\frac{3}{2}\): \(m_p = -\frac{2}{3}\) So, the slope of the perpendicular line is \(-\frac{2}{3}\).

Find the y-intercept of the perpendicular line

Since the perpendicular line passes through the point (2, -4), we'll plug these values into the equation \(y = mx + b\) to find its y-intercept: -4 = -\frac{2}{3} \cdot 2 + b -4 = -\frac{4}{3} + b Now, add \(\frac{4}{3}\) to both sides to find the value of \(b\): b = -4 + \frac{4}{3} = -\frac{8}{3}

Write the equation of the perpendicular line

Now that we have the slope and y-intercept of the perpendicular line, we can write its equation using the slope-intercept form (\(y = mx + b\)): y = -\frac{2}{3}x - \frac{8}{3} So, the equation of the line passing through the point (2, -4) and perpendicular to the line \(3x - 2y = 6\) is \(y = -\frac{2}{3}x - \frac{8}{3}\).

Key concepts

Slope-Intercept Form

The slope-intercept form is a way to express the equation of a line using the formula \(y = mx + b\). This form is particularly handy when working with lines, since it clearly shows both the slope \(m\) and the y-intercept \(b\) of the line.
- **Slope** \(m\) represents the steepness of the line and its direction. Positive slope means the line goes upwards, while negative means it goes downwards.
- **Y-intercept** \(b\) is the point where the line crosses the y-axis. This is where \(x = 0\).
In our problem, the perpendicular line's equation \(y = -\frac{2}{3}x - \frac{8}{3}\) uses the slope-intercept form. Here, \( -\frac{2}{3} \) is the slope and \( -\frac{8}{3} \) is the y-intercept.

Point-Slope Formula

The point-slope formula is another form to write the equation of a line. It is useful when you know a point on the line and its slope. The formula is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a known point on the line, and \(m\) is the slope.
To find the equation of our perpendicular line, we used the fact that it passes through the point \((2, -4)\) and has a slope \(-\frac{2}{3}\). Plugging these values into the point-slope formula, we have:
\[ y - (-4) = -\frac{2}{3}(x - 2) \]
Which simplifies to:
\[ y + 4 = -\frac{2}{3}x + \frac{4}{3} \]
Rearranging gives us the slope-intercept form we found earlier: \[ y = -\frac{2}{3}x - \frac{8}{3} \]

Line Equation

A line equation is a mathematical expression describing all the points along a line. There are multiple ways to express this, with the slope-intercept and point-slope forms being the most common. Knowing these can help in transitioning between different equation forms and understanding the relationship between line characteristics.
Every line in the plane can be described by an equation. Key forms include:
- **Standard Form**: \(Ax + By = C\)
- **Slope-Intercept Form**: \(y = mx + b\)
These equations let you determine properties like slope and y-intercept at a glance. In our original exercise, converting the given line to slope-intercept form showed us its slope \(\frac{3}{2}\), which helped find the perpendicular line's slope as \(-\frac{2}{3}\).
Using these forms allows us to easily write the equation for any line, whether given a point and slope or two points. As a result, understanding the line equation concept is crucial for navigating between different forms and solving problems involving straight lines.