Find the equation of the line perpendicular to 5 x − 3 y + 4

To find the equation of the line perpendicular to $5x - 3y + 4 = 0$ and passing through the point $(-4, 1)$, follow these steps:

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Step 1: Find the slope of the given line.

Rewriting $5x - 3y + 4 = 0$ in slope-intercept form ($y = mx + b$): $5x - 3y + 4 = 0 \quad \Rightarrow \quad -3y = -5x - 4 \quad \Rightarrow \quad y = \frac{5}{3}x + \frac{4}{3}.$ The slope of the given line is $m = \frac{5}{3}$.

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Step 2: Determine the slope of the perpendicular line.

The slope of a line perpendicular to another is the negative reciprocal of the original slope. Hence, the slope of the perpendicular line is: $m_{\text{perpendicular}} = -\frac{1}{m} = -\frac{3}{5}.$

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Step 3: Use the point-slope form of the line equation.

The point-slope form is given by: $y - y_1 = m(x - x_1),$ where $m = -\frac{3}{5}$ and $(x_1, y_1) = (-4, 1)$.

Substitute these values: $y - 1 = -\frac{3}{5}(x + 4).$

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Step 4: Simplify the equation.

Distribute the slope on the right-hand side: $y - 1 = -\frac{3}{5}x - \frac{12}{5}.$ Add 1 to both sides ($1 = \frac{5}{5}$): $y = -\frac{3}{5}x - \frac{12}{5} + \frac{5}{5}.$ Simplify: $y = -\frac{3}{5}x - \frac{7}{5}.$

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Final Answer:

The equation of the perpendicular line is: $y = -\frac{3}{5}x - \frac{7}{5}.$

Would you like further clarification or additional details?

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Related Questions:

1. How do you find the equation of a line parallel to another line?
2. What does the negative reciprocal of a slope represent geometrically?
3. Can you find the equation of a line given two points?
4. How do you convert a line equation from slope-intercept to standard form?
5. What is the significance of perpendicular slopes in coordinate geometry?

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Tip:

When working with perpendicular lines, always check the product of their slopes—it should equal $-1$.