How do you use the point on the line and the slope of the line to find three additional points through which the line passes: Point: (7, -2) Slope:m = 1/2?

Question

Isaiah Anthony · Accepted Answer

Here's how you can do that.

All you have to know is that the line's slope contains a set of directions that let you locate additional points that are on the same line by starting from a point that is on the line. Hence, you are aware that the slope of a certain line is #m = 1/2# As you know, the slope of a line is defined as the change in #y#, or #Deltay#, divided by the change in #x#, or #Deltax# #m = (Deltay)/(Deltax)# Now, you know that the point #(7,-2)# lies on this line. The change in #y# tells you the number of positions that you must move up on the #y# axis in order to find the #y#-coordinate of another point that lies on the line. Similarly, the change in #x# tells you the number of positions that you must move to the right on the #x# axis in order to find the #x# coordinate of another point that lies on the line. Here, you've #m = 1/2 implies {(Deltay = 1), (Deltax = 2) :}# So, if you start at #x=7#, you must move #2# positions to the right to find #x_2 = 7 + 2 = 9# Similarly, if you start at #y=-2#, you mus move #1# position up to find #y_2 = -2 + 1 = -1# Therefore, a second point on the given line is #(9,-1)#. Now here comes the cool part, You can use multiples of the slope to find additional points by starting from the same point #(7,-2)#. For example, you have #m = 1/2 = 2/4# This implies that you will receive #{(x_3 = 7 + 4 = 11), (y_3 = -2 + 2 = 0) :} implies (11,0)# is another point that lies on the line In the same way, you can also have #m = 1/2 = (-1)/(-2)# In this case, you're moving #2# positions to the left for #x# and #1# position down for #y#. Thus, it follows that #{(x_4 = 7 + (-2) = 5), (y_4 = -2 + (-1) = -3) :} implies (5,-3)# is another point that lies on the line Therefore, you can say that #(5,-3)#, #(7,-2)#, #(9,-1)#, and #(11,0)# are all points that lie on the given line. Use one of the points to write the line's equation so you can verify the outcome twice. #(y - y_4) = m * (x - x_4)# #y - 0 = 1/2 * (x - 11)# #y = 1/2x - 11/2# This is how the line appears. graph{1/2x - 11/2 [-5, 5, 10, -10]} As you can see, every point we were able to locate is on the border.

Avery Alexander · Answer

undefined To find three additional points through which the line passes given a point on the line (7, -2) and the slope (m = 1/2), you can use the slope-intercept form of a linear equation: y = mx + b.

1. Substitute the given point's coordinates (7, -2) into the equation to solve for the y-intercept (b).
2. Once you find the y-intercept, you will have the equation of the line.
3. Use the slope (m) and y-intercept (b) to find three additional points by incrementing or decrementing the x-coordinate from the given point and then using the equation to find the corresponding y-coordinate.

For example:
- Start with the given point (7, -2).
- Use the slope-intercept form to find the y-intercept: y = (1/2)x + b.
- Substitute the given point (7, -2) into the equation: -2 = (1/2)(7) + b.
- Solve for b to find the y-intercept.
- Once you have the equation of the line, increment or decrement the x-coordinate by a certain value (e.g., 1) and use the equation to find the corresponding y-coordinate.
- Repeat this process to find three additional points on the line.