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