Find the equation of a line perpendicular to line 2y = 5x + 4 which passes through (4, 2).

1. Analyze the question

We need to find the equation of a line that is perpendicular to a given line ($2y = 5x + 4$) and passes through a specific point $(4, 2)$.

2. Explain the concept

• Slope-Intercept Form: A line equation in the form $y = mx + c$ has a slope of $m$.
• Perpendicular Slopes: If two lines are perpendicular, the product of their slopes is $-1$. That is, $m_1 \times m_2 = -1$, or $m_2 = -\frac{1}{m_1}$.
• Point-Slope Formula: To find the equation of a line passing through $(x_1, y_1)$ with slope $m$, we use: $y - y_1 = m(x - x_1)$.

3. Show the solution

Step 1: Find the slope of the given line. The given equation is $2y = 5x + 4$. Divide by 2 to get it into $y = mx + c$ form: $y = \frac{5}{2}x + 2$ The slope ($m_1$) is $\frac{5}{2}$.

Step 2: Find the slope of the perpendicular line ($m_2$). Since $m_1 \times m_2 = -1$: $\frac{5}{2} \times m_2 = -1$ $m_2 = -\frac{2}{5}$

Step 3: Use the point-slope formula with point (4, 2). $y - y_1 = m_2(x - x_1)$ $y - 2 = -\frac{2}{5}(x - 4)$

Step 4: Simplify the equation. Multiply both sides by 5 to clear the fraction: $5(y - 2) = -2(x - 4)$ $5y - 10 = -2x + 8$

Rearrange to match the format of the options (bringing all terms to one side): $5y + 2x - 10 - 8 = 0$ $5y + 2x - 18 = 0$

4. Identify the answer

Comparing our result to the given options, the correct equation is $5y + 2x - 18 = 0$. This matches option B.

5. Tip

When dealing with perpendicular lines, a quick trick to find the new slope is to "flip and negate" the original slope. If the original slope is $\frac{a}{b}$, the perpendicular slope is $-\frac{b}{a}$. Also, remember that for any line $Ax + By + C = 0$, a perpendicular line will always take the form $Bx - Ay + K = 0$.