what is the distance between origin and solution of equation x+3y=6 and 2x-3y=12
Answers
2.5 units (approximately)
Step-by-step explanation:
1
coordinate axis with origin at (0,0) as 2x + 3y = 6 and 2x + 3y = 12. A new road equidistant from both is built. What is the minimum distance from origin to the new road?
Lauri Barber
Answered March 1, 2018
Put each equation into slope-intercept form of y = mx + b by subtracting 2x from both sides; then dividing ALL terms by 3: y = -2/3x + 2 and y = -2/3x + 4.
The first line would go through the y-axis at 2 while the second line would go through 4. So an equidistant line would go through 3 on the y-axis. They all would have the same slope of -2/3.
Draw a graph of these 3 lines, each going down two and right 3 spaces to find a second point for each line. First line: Go down 2 from the y-intercept 2 to 0 and right 3 to the x value of 3. The new point would be (3,0). Second line: Go down 2 from the y-intercept of 4 to 2 and right 3 to x-value of 3. The new point is (3,2). Middle line: Go down 2 from the y-intercept of 3 to 1 and right 3 to 3. The new point for this line is (3,1).
NOTE: There are three parallel lines going down to the right. The distance to the origin from the y-axis = 3. If you want to know where the newest line crosses the x-axis, calculate 0 = -2/3x + 3 and you get 4.5. This is farther from the origin than 3. Looking at the graph, I think that the perpendicular line going to the line defined as y = -2/3x + 3 would be closest.
The slope of a perpendicular line is the opposite sign and reciprocal of -2/3, which is a slope of 3/2. Since it also goes through the origin, the y-intercept is 0, so the equation for this line is: y = 3/2x. The point where y = 3/2x and y = -2/3x +3 are equal is the intersection point.
3/2x = -2/3x + 3 → x = 18/13 or 1 and 5/13
3/2 (18/13) = y →y = 27/13 or 2 and 1/13
The coordinates for the closest point (18/13, 27/13). Use the Distance Formula for this point and the Origin to find the distance: SqRt [(18/13 - 0)^2 + (27/13 - 0)^2] → SqRt [1.917159763 + 4.313609467] = SqRt[6.23] = 2.495996795 or approximately 2.5 units from the Origin.