Find the equation of straight line passing through (2, 3) and perpendicular to the line $3x + 2y + 4 = 0$

1. Analyze the Question

We need to find the equation of a line that passes through a specific point $(2, 3)$ and is perpendicular to another line given by the equation $3x + 2y + 4 = 0$.

2. Explain the Concept

• Slope-Intercept Form: Any linear equation can be written as $y = mx + c$, where $m$ is the slope.
• Perpendicular Lines: If two lines are perpendicular, the product of their slopes is $-1$. That is, $m_1 \times m_2 = -1$. This means the slope of the new line ($m_2$) is the negative reciprocal of the original slope ($m_1$).
• Point-Slope Formula: Once we have the slope ($m$) and a point $(x_1, y_1)$, we use the formula: $y - y_1 = m(x - x_1)$.

3. Show the Solution

Step 1: Find the slope of the given line. The given equation is $3x + 2y + 4 = 0$. Let's rearrange it into $y = mx + c$ form: $2y = -3x - 4$ $y = -\frac{3}{2}x - 2$ The slope ($m_1$) of the given line is $-\frac{3}{2}$.

Step 2: Find the slope of the perpendicular line ($m_2$). Since the lines are perpendicular: $m_1 \times m_2 = -1$ $(-\frac{3}{2}) \times m_2 = -1$ $m_2 = \frac{-1}{-3/2} = \frac{2}{3}$

Step 3: Use the point-slope formula with point (2, 3). $y - y_1 = m(x - x_1)$ $y - 3 = \frac{2}{3}(x - 2)$

Multiply both sides by 3 to clear the fraction: $3(y - 3) = 2(x - 2)$ $3y - 9 = 2x - 4$

Rearrange to match the format of the options: $3y = 2x - 4 + 9$ $3y = 2x + 5$

4. Identify the Answer

Comparing our result ($3y = 2x + 5$) with the given options, it matches option D.

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5. Exam Tip

When dealing with perpendicular lines, a quick trick to find the new equation is to swap the coefficients of $x$ and $y$ and change the sign between them. For $3x + 2y + 4 = 0$, swapping the coefficients gives $2x - 3y + k = 0$. Plug in the point $(2, 3)$ to find $k$: $2(2) - 3(3) + k = 0 \Rightarrow 4 - 9 + k = 0 \Rightarrow k = 5$. Thus, $2x - 3y + 5 = 0$, which rearranges to $3y = 2x + 5$.