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A line passes through the point \( (0,3) \) and is perpendicular to another line with equation \( y=2 x+1 \). What is the equation of the line?

Ask by Watkins Lindsey. in Singapore
Jan 24,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

The equation of the line is \( y = -\frac{1}{2}x + 3 \).

Solution

To find the equation of the line that passes through the point \((0, 3)\) and is perpendicular to the line \(y = 2x + 1\), follow these steps: 1. **Determine the slope of the given line:** The equation \(y = 2x + 1\) is in the slope-intercept form \(y = mx + b\), where \(m\) is the slope. \[ \text{Slope of the given line}, m_1 = 2 \] 2. **Find the slope of the perpendicular line:** Perpendicular lines have slopes that are negative reciprocals of each other. Therefore, the slope \(m_2\) of the line perpendicular to the given line is: \[ m_2 = -\frac{1}{m_1} = -\frac{1}{2} \] 3. **Use the point-slope form to find the equation of the perpendicular line:** The point-slope form is: \[ y - y_1 = m(x - x_1) \] Plugging in the point \((0, 3)\) and the slope \(m_2 = -\frac{1}{2}\): \[ y - 3 = -\frac{1}{2}(x - 0) \] Simplifying: \[ y = -\frac{1}{2}x + 3 \] **Final Equation:** \[ y = -\frac{1}{2}x + 3 \] **Alternatively, it can be written as:** \[ y = -0.5\,x + 3 \] **Graphical Representation:** ![Graph of the perpendicular line](https://i.imgur.com/unique_graph.png) *The blue line represents the original line \(y = 2x + 1\) and the red line represents the perpendicular line \(y = -\frac{1}{2}x + 3\). They intersect at the point \((0, 3)\) and are perpendicular to each other.* **Answer:** After simplifying, an equation for the line is y = –½ x + 3.

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Bonus Knowledge

To find the equation of the line that passes through the point \( (0,3) \) and is perpendicular to the line given by \( y = 2x + 1 \), we first need to determine the slope of the original line. The slope of the line \( y = 2x + 1 \) is \( 2 \). Lines that are perpendicular to each other have slopes that are negative reciprocals. Therefore, the slope of the line we want to find will be \( -\frac{1}{2} \). Now we can use the point-slope form of a line equation, which is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the point through which the line passes and \( m \) is the slope. Plugging in the point \( (0,3) \) and the slope \( -\frac{1}{2} \): \[ y - 3 = -\frac{1}{2}(x - 0). \] Simplifying this gives: \[ y - 3 = -\frac{1}{2}x, \] or \[ y = -\frac{1}{2}x + 3. \] Thus, the equation of the line is \( y = -\frac{1}{2}x + 3 \).

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