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\begin{tabular}{c} Write the equation of \\ \( 4 \begin{array}{c}\text { the line perpendicular to } \\ y=5 x-1 \text { through }(15,3)\end{array} \) \\ \hline\end{tabular}

Ask by Cruz Turner. in the United States
Feb 04,2025

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The equation of the line perpendicular to \( y = 5x - 1 \) and passing through \( (15, 3) \) is \( y = -\frac{1}{5}x + 6 \).

Solución

To find the equation of the line that is perpendicular to \( y = 5x - 1 \) and passes through the point \( (15, 3) \), follow these steps: 1. **Determine the slope of the given line:** The equation \( y = 5x - 1 \) is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope. So, the slope \( m_1 \) of the given line is: \[ m_1 = 5 \] 2. **Find the slope of the perpendicular line:** The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore, the slope \( m_2 \) of the perpendicular line is: \[ m_2 = -\frac{1}{m_1} = -\frac{1}{5} \] 3. **Use the point-slope form to write the equation of the perpendicular line:** The point-slope form is: \[ y - y_1 = m(x - x_1) \] Plugging in the slope \( m_2 = -\frac{1}{5} \) and the point \( (15, 3) \): \[ y - 3 = -\frac{1}{5}(x - 15) \] 4. **Simplify to the slope-intercept form:** Distribute the slope and solve for \( y \): \[ y - 3 = -\frac{1}{5}x + 3 \] \[ y = -\frac{1}{5}x + 6 \] **Final Equation:** \[ y = -\frac{1}{5}x + 6 \]

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Bonus Knowledge

To find the equation of the line that is perpendicular to \( y = 5x - 1 \) and passes through the point \( (15, 3) \), we first determine the slope of the given line, which is 5. The slope of the line perpendicular to this will be the negative reciprocal, giving us a slope of \( -\frac{1}{5} \). Using the point-slope form of the equation \( y - y_1 = m(x - x_1) \), we substitute the points and slope: \[ y - 3 = -\frac{1}{5}(x - 15). \] Now, simplifying: \[ y - 3 = -\frac{1}{5}x + 3, \] \[ y = -\frac{1}{5}x + 6. \] So, the equation of the line is \( y = -\frac{1}{5}x + 6 \). If you're looking for more depth, understanding the concept of slopes in coordinate geometry can add to your appreciation for lines! Remember, perpendicular lines not only create right angles but can also intersect at multiple points, creating fascinating patterns in geometry. Want an exciting math challenge? Try graphing both lines on the same coordinate plane! You’ll see how they beautifully intersect and the significance of their slopes. Who knows, you might even discover more lines that pass through the same point or analyze their intercepts!

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