Answers for Determine The Minimum Distance From The Point (18,20) To The Line 5x+8y=120 Include: Graph, Algebra Of Any Equations, Algebra Of System, Set-up & Steps Of Final Calculations.

The distance from point (a,b) to the line

Ax + By + C = 0
is
|Aa + Bb + C| / √(A^2 + B^2)

write the line as 5x + 8y - 120 = 0 and plug in values

To determine the minimum distance from the point (18, 20) to the line 5x + 8y = 120, we can use the following steps:

Step 1: Graph the line and the point on a coordinate plane:
- Plot the point (18, 20) on the coordinate plane.
- To graph the line 5x + 8y = 120, we can rearrange the equation to slope-intercept form (y = mx + b) by solving for y:
8y = -5x + 120
y = (-5/8)x + 15
- Plot some additional points on the line and connect them to form a straight line.

Step 2: Determine the slope of the line:
- The slope-intercept form of the equation (y = mx + b) tells us that the coefficient of x (-5/8 in this case) is the slope of the line.

Step 3: Set up a system of equations:
- Let's assume that there is a point (x, y) on the line that is closest to the point (18, 20).
- The distance between two points can be calculated using the distance formula: D = √[(x2 - x1)^2 + (y2 - y1)^2].
- We can express the distance between the two points as a function of x and solve for the minimum value of D.
- So, the distance (D) is equal to √[(x - 18)^2 + (y - 20)^2].

Step 4: Express y in terms of x:
- Since the point (x, y) is on the line, we can substitute the value of y from the equation of the line:
y = (-5/8)x + 15
- Substitute this expression into the distance formula for the value of y.

Step 5: Simplify the distance formula:
- Substitute the expression for y and simplify the distance formula, D = √[(x - 18)^2 + ((-5/8)x + 15 - 20)^2].

Step 6: Minimize the distance formula:
- To find the minimum distance, we need to minimize the distance formula. This can be done by finding the x-value that minimizes the expression inside the square root.
- Take the derivative of the expression inside the square root with respect to x, and set it equal to zero to find the critical points.

Step 7: Solve for x:
- Solve the equation obtained in the previous step to find the x-coordinate of the point that minimizes the distance.

Step 8: Calculate the y-coordinate and the minimum distance:
- Substitute the value of x into the equation of the line to find the corresponding y-coordinate.
- Substitute both x and y values into the distance formula to calculate the minimum distance.

Step 9: Interpret the result:
- Once you have the minimum distance, you can interpret it in the context of the problem.

By following these steps, you should be able to determine the minimum distance from the point (18, 20) to the line 5x + 8y = 120.