Questions: Find the equation of the line that passes through (-2,3) and is perpendicular to the line passing through (-6, 1/3) and (-4, 2/5). Write the equation in slope-intercept form. The equation is y= (Simplify your answer. Use integers or fractions for any numbers in the expression. Do not factor.)

Solution Steps

To find the equation of the line that is perpendicular to another line, we first need to determine the slope of the given line. The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as \((y_2 - y_1) / (x_2 - x_1)\). Once we have the slope of the given line, the slope of the perpendicular line is the negative reciprocal of this slope. With the slope of the perpendicular line and a point it passes through, we can use the point-slope form of a line equation to find the equation in slope-intercept form.

To find the slope of the line passing through the points \((-6, \frac{1}{3})\) and \((-4, \frac{2}{5})\), we use the formula for the slope:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\frac{2}{5} - \frac{1}{3}}{-4 - (-6)} \]

Calculating the difference in the \(y\)-coordinates:

\[ \frac{2}{5} - \frac{1}{3} = \frac{6}{15} - \frac{5}{15} = \frac{1}{15} \]

Calculating the difference in the \(x\)-coordinates:

\[ -4 - (-6) = 2 \]

Thus, the slope of the given line is:

\[ m = \frac{\frac{1}{15}}{2} = \frac{1}{30} \]

The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore, the slope of the perpendicular line is:

\[ m_{\perp} = -\frac{1}{\frac{1}{30}} = -30 \]

We use the point-slope form of a line equation, \(y - y_1 = m(x - x_1)\), with the point \((-2, 3)\) and the slope \(-30\):

\[ y - 3 = -30(x + 2) \]

Expanding this equation:

\[ y - 3 = -30x - 60 \]

Solving for \(y\) to get the equation in slope-intercept form:

\[ y = -30x - 60 + 3 \]

\[ y = -30x - 57 \]

Final Answer