The equation of the line that is perpendicular to $y=4x+5$ and contains the point $(8,-4)$ is
$y=-\frac{[?]}{[]} x+[]$
Answer (1)
Determine the slope of the perpendicular line: Since the given line has a slope of 4, the perpendicular line has a slope of − 4 1 .
Use the point-slope form: Substitute the point ( 8 , − 4 ) and the slope − 4 1 into the point-slope form y − y 1 = m ( x − x 1 ) .
Simplify the equation: Simplify y − ( − 4 ) = − 4 1 ( x − 8 ) to slope-intercept form.
State the final equation: The equation of the perpendicular line is y = − 4 1 x − 2 , so the answer is y = − 4 1 x − 2 , which means the missing values are 1, 4, and -2. y = − 4 1 x − 2
Explanation
Understanding the Problem The problem asks us to find the equation of a line that is perpendicular to the line y = 4 x + 5 a and passes through the point ( 8 , − 4 ) . We need to express the equation in the form y = − [ ] [ ?] x + [ ] .
Finding the Slope of the Perpendicular Line The given line has a slope of 4. A line perpendicular to this line will have a slope that is the negative reciprocal of 4. Therefore, the slope of the perpendicular line is − 4 1 .
Using the Point-Slope Form Now we use the point-slope form of a line, which is y − y 1 = m ( x − x 1 ) , where ( x 1 , y 1 ) is the given point ( 8 , − 4 ) and m is the slope − 4 1 .
Substituting the Values Substitute the values into the point-slope form: y − ( − 4 ) = − 4 1 ( x − 8 ) .
Simplifying the Equation Simplify the equation to slope-intercept form, y = m x + b . First, we have y + 4 = − 4 1 x + 2 .
Solving for y Subtract 4 from both sides to solve for y : y = − 4 1 x + 2 − 4 , which simplifies to y = − 4 1 x − 2 .
Final Answer The equation of the perpendicular line is y = − 4 1 x − 2 . Thus, the missing values are 1, 4, and -2.
Examples
Understanding perpendicular lines is crucial in various real-world applications, such as architecture and navigation. For instance, architects use perpendicular lines to design stable and aesthetically pleasing structures, ensuring walls meet floors at right angles. In navigation, understanding perpendicular paths helps ships and airplanes maintain safe distances and avoid collisions. This concept also applies to city planning, where streets are often laid out in a grid pattern, forming perpendicular intersections to optimize traffic flow and accessibility.
