5. \( x+y=3 ;(5,4) \) Write an equation of the line that is perpendicular to the given line and through the given point. \( \begin{array}{ll}\text { 7. } y=-4 x+2 y=1 ;(-2,6, \\ \text { 8. } y=\frac{2}{3} x+6 ;(-6,2) & \text { 7. Wy }\end{array} \)

To solve the problem, we need to find the equation of a line that is perpendicular to the given line \( x + y = 3 \) and passes through the point \( (5, 4) \). ### Step 1: Find the slope of the given line The equation of the line is given in standard form \( Ax + By = C \). We can convert it to slope-intercept form \( y = mx + b \) to find the slope. Starting with the equation: \[ x + y = 3 \] We can rearrange it to solve for \( y \): \[ y = -x + 3 \] From this, we see that the slope \( m_1 \) of the given line is \( -1 \). ### Step 2: Find the slope of the perpendicular line The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, if the slope of the given line is \( m_1 = -1 \), the slope \( m_2 \) of the perpendicular line is: \[ m_2 = -\frac{1}{m_1} = -\frac{1}{-1} = 1 \] ### Step 3: Use the point-slope form to find the equation of the perpendicular line The point-slope form of a line is given by: \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope. Using the point \( (5, 4) \) and the slope \( m_2 = 1 \): \[ y - 4 = 1(x - 5) \] ### Step 4: Simplify the equation Now, we can simplify this equation: \[ y - 4 = x - 5 \] \[ y = x - 1 \] ### Final Answer The equation of the line that is perpendicular to \( x + y = 3 \) and passes through the point \( (5, 4) \) is: \[ y = x - 1 \]