Find the equation of the line that is perpendicular to 𝑦=23𝑥+1 and passes through the point (7,5)

1. Break Down the Problem

To find the equation of a line that is perpendicular to a given line, we need to:

1. Find the slope of the given line.
2. Determine the slope of the perpendicular line.
3. Use the point-slope form to write the equation of the new line using the point (7, 5).

2. Relevant Concepts

The slope-intercept form of a line is given by: $y = mx + b$ where $m$ is the slope. For two lines to be perpendicular, the product of their slopes must equal -1: $m_1 \cdot m_2 = -1$ Thus, if we know the slope $m_1$ of the original line, the slope $m_2$ of the perpendicular line can be calculated as: $m_2 = -\frac{1}{m_1}$

3. Analysis and Detail

1. The equation of the given line is: $y = 23x + 1$ Here, the slope $m_1 = 23$.

2. For the perpendicular slope, we calculate: $m_2 = -\frac{1}{23}$

3. Using the point-slope form $y - y_1 = m(x - x_1)$ where $(x_1, y_1) = (7, 5)$ and $m = -\frac{1}{23}$: $y - 5 = -\frac{1}{23}(x - 7)$

4. Verify and Summarize

Now we simplify the equation: $y - 5 = -\frac{1}{23}x + \frac{7}{23}$ Combining terms: $y = -\frac{1}{23}x + 5 + \frac{7}{23}$ We convert 5 to a fraction: $5 = \frac{115}{23}$ So: $y = -\frac{1}{23}x + \frac{115}{23} + \frac{7}{23}$ $y = -\frac{1}{23}x + \frac{122}{23}$

Final Answer

The equation of the line that is perpendicular to $y = 23x + 1$ and passes through the point (7, 5) is: $y = -\frac{1}{23}x + \frac{122}{23}$