Questions: Write an equation of the line containing the given point and perpendicular to the given line. Express your answer in the form y=mx+b. (4,5) ; 5x+y=6 The equation of the line is y= (Simplify your answer. Use integers or fractions for any numbers in the expression.)

Solution Steps

To find the equation of the line that is perpendicular to the given line and passes through the given point, we first need to determine the slope of the given line. The given line is in the form \(5x + y = 6\), which can be rewritten in slope-intercept form \(y = mx + b\). The slope of the line perpendicular to this will be the negative reciprocal of the slope of the given line. Once we have the slope of the perpendicular line, we can use the point-slope form of a line equation to find the equation of the line that passes through the given point \((4, 5)\).

The given line is represented by the equation \(5x + y = 6\). We can rewrite this in slope-intercept form \(y = mx + b\): \[ y = 6 - 5x \] From this, we identify the slope \(m\) of the given line as \(-5\).

The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. Therefore, the slope \(m_{\text{perpendicular}}\) is: \[ m_{\text{perpendicular}} = -\frac{1}{m} = -\frac{1}{-5} = \frac{1}{5} \]

We will use the point-slope form of the line equation, which is given by: \[ y - y_1 = m(x - x_1) \] Substituting the point \((4, 5)\) and the slope \(\frac{1}{5}\): \[ y - 5 = \frac{1}{5}(x - 4) \] Simplifying this, we get: \[ y - 5 = \frac{1}{5}x - \frac{4}{5} \] Adding \(5\) to both sides: \[ y = \frac{1}{5}x + 5 - \frac{4}{5} = \frac{1}{5}x + \frac{25}{5} - \frac{4}{5} = \frac{1}{5}x + \frac{21}{5} \]

Final Answer