Questions: Write an equation of the line satisfying the given conditions. Write the answer in slope-intercept form or simplified fractions. The line is perpendicular to 7y+2x=49 and passes through the point (2, 4). The equation of the line is .

Solution Steps

To find the equation of a line that is perpendicular to a given line and passes through a specific point, follow these steps:

1. Find the slope of the given line: Rewrite the equation in slope-intercept form \(y = mx + b\) to identify the slope \(m\).
2. Determine the perpendicular slope: The slope of a line perpendicular to another is the negative reciprocal of the original slope.
3. Use the point-slope form: With the perpendicular slope and the given point, use the point-slope form of a line equation \(y - y_1 = m(x - x_1)\) to find the equation of the line.
4. Convert to slope-intercept form: Simplify the equation to get it into the form \(y = mx + b\).

The given line is represented by the equation \(2x + 7y = 49\).

Rearranging the equation into slope-intercept form \(y = mx + b\), we find the slope \(m\) of the given line: \[ m = -\frac{2}{7} \]

The slope of the line that is perpendicular to the given line is the negative reciprocal of \(-\frac{2}{7}\): \[ m_{\text{perpendicular}} = \frac{7}{2} \]

Using the point \((2, 4)\) and the perpendicular slope \(\frac{7}{2}\), we apply the point-slope form: \[ y - 4 = \frac{7}{2}(x - 2) \]

Expanding and simplifying the equation gives: \[ y - 4 = \frac{7}{2}x - 7 \] \[ y = \frac{7}{2}x - 3 \]

Final Answer