findth e im ag e o f th eli n e. y=3x+4.after.reflectioneintheliney=2x -3?
Y=91\63x -434\63
This article will probably help confirm your answer
http://www.sdmath.com/math/geometry/reflection_across_line.html
I plotted your 3 lines.
Looks like your slope is off, but your common point looks correct
www.wolframalpha.com/input/?i=plot+y%3D3x%2B4,+y%3D2x+-3+,+y%3D91%2F63x+-434%2F63
suppose you want to reflect the line L2:y=cx+d across the line L1:y=ax+b,
creating a new line L3: y=mx+n
You know they intersect at some point (h,k), in this case, (-7,-17)
Consider the slopes of L1 and L2. They form angles A and C with the x-axis, such that
tanA = a
tanC = c
Our new line L3 will form an angle M with the x-axis, so that tanM = m
Now, we want to find M such that C-A = A-M
That is, M = 2A-C
m = tanM = tan(2A-C)
= (tan2A -tanC)/(1 + tanC tan2A)
= (tan2A - c)/(1 + c*tan2A)
Now, tan2A = 2a/(1-a^2), so
m = ([2a/(1-a^2)] - c)/(1+c[2a/(1-a^2)])
m = -(a^2c + 2a - c)/(a^2 - 2ac - 1)
For our problem, a=2 and c=3, so
m = 13/9
Using the point-slope form of the line, our reflected line L3 has equation
y+17 = 13/9 (x+7)
To find the image of the line y = 3x + 4 after reflection in the line y = 2x - 3, we need to consider the properties of reflection.
The reflection of a point (x, y) in a line can be found by following these steps:
1. Calculate the slope of the line y = 2x - 3.
- We can see that the slope of this line is 2.
2. Use the formula to find the perpendicular slope.
- Perpendicular slope = -1 / original slope
- So, the perpendicular slope = -1 / 2 = -0.5
3. Calculate the negative reciprocal of the original slope to find the perpendicular slope.
- The negative reciprocal of 2 is -0.5.
4. Identify the midpoint between the two lines.
- In this case, the midpoint can be calculated by finding the solution to the system of equations y = 3x + 4 and y = 2x - 3.
- By setting the two equations equal to each other, we can solve for the x-coordinate: 3x + 4 = 2x - 3.
- Solving this equation gives us x = -7.
5. Substitute the midpoint (x-value) into the line equation y = 3x + 4 to find the corresponding y-value.
- Plugging in x = -7, we get y = 3(-7) + 4 = -17.
6. Calculate the distance between the midpoint and any point on the original line.
- Taking the distance from the midpoint (-7, -17) to any point on the original line y = 3x + 4 will give us the distance between the two lines.
By following these steps, we can find the image of the line y = 3x + 4 after reflection in the line y = 2x - 3 and determine its properties.