Write the equation of the line in slope-intercept form that passes through the point (– 9, 12) and is perpendicular to the line passing through the points (6, - 2) and (- 19, 3).

Answer 1

Hi J T.,

My name is Andrew. To answer this question I would start by asking how comfortable you are with finding this answer. If you are not comfortable graphing then I would look at the following description. If want to answer this question using only numbers and no graph you can jump to "Using just numbers Math:"

Graphing:

if you have a piece of graph paper that may be the easiest. If you do not have graph paper that is fine as well. I would start by making the X Y axis. then I would place a dash line on the x axis every 5 points starting at 0 going to 5 then to 10 then to 15 and then finally to 20. Use a ruler or something that is hard to make the axis straight lines. I would do this for the negative x axis as well going to -5, -10, -15, and then -20. You can then make a Y axis in a similar fashion placing marks on the axis every 5 points.

Here is my guide to help place points on the graph:

The next step is to plot the points on the graph, if you need this to be exact then using a ruler or hard surface is needed. This question is asking that a line needs to go through the points (6,-2) and through (-19,3). In math when a point is described the parentheses are used to close the x and y point. The first number is the x point and the second number that is after the comma is the y point. I would start with placing a point where you think 6 may be on the x axis, then I would go down the y axis 2 points, this is where the negative 2 is located, I would make marks on the axis and then I would line up the two points you made on the axis and make a guess where (6,-2) is. I would follow the same steps for (-19,3) but now for the X point you move to the left and the Y point you move up. Then you can use the ruler to draw a line between the two points. Follow the same steps for the third point.

Here is my guide for understanding the line between the points:

Once the two points (-19,3) and (6,-2) are placed on the graph, draw a line between the two points. This is where graph paper can be helpful to visually see the next part, if graph paper was not used you are still ok! Now a line between two points has several things that we can interpret from this line. The first thing we can make is a triangle from the line and the two points (-19,3) and (6,-2). Using a ruler or other object draw a line that is going down at the point (-19,3). Then make a line going to the left of the point (6,-2) and continue this point so that the up and down line you drew connects with this line to the left of (6,-2). I hope now that this new shape should look like a triangle.

You can now count the number of "spaces" up and down if you have graph paper. If you do not have graph paper then if you find the distance between the (....,Y's) this will be the side of this triangle or the "rise" of slope of the line. The points are 3 minus -2. This is 5, so the side of the triangle is 5. Then the "run" of the line is the bottom part of the triangle and is 6 minus -19 which is 25.

Rise over Run:

So the left side of the triangle is the "rise" and should be 5, and the "run" of the triangle should be 25. These two numbers are used for determining where a parallel line will go through the point (-9,12). So The other note about this line is that if you trace your finger from left to right the line is going down. This means the slope is negative. Also, the slope is made by taking the rise/run and turning that number negative. So the slope should be -5/25 and if you would like you can simplify this to be -1/5.

so all we need to do to make a line that has the same slope that we made using triangle. I interpret the rise to be -1 meaning you go down from the point we want at (-9,12) 1 point, then you "run" to the right 5 points. Make a new point on the graph and do this again going down 1 point and over to the right 5 points. With three points on the graph you can make the line extend through the axis. Here you may not be able to use the axis to get the intercept. but you should be able to get an approximate that will match the numbers way for finding this answer.

Using just numbers and no Graph:

The equation of a line for the slope is m = (y₂ - y₁) / (x₂ - x₁)

so using the two points (-19,3) and (6,-2) m = (3 - (-2)) / ((-19) - 6)

turns into m = (3+2) / (-25)

m = - 5 / 25

or m = -1 / 5

notice this method is a check for finding the slope using the triangle "rise" "run" method.

Then slope of a line is: y - y₁ = m ( x - x₁)

so picking the point we care about at (-9,12)

y - 12 = (-1 / 5) (x - (-9))

y - 12 = (-1/5) (x + 9)

y - 12 = (-1/5)x - (1/5) (9)

y - 12 = (-1/5)x - 9/5

y = (-1/5)x - 9/5 + 12

y = (-1/5)x - 9/5 + 12 (5 / 5)

y = (-1/5)x - 9/5 + 60/5

y = (-1/5)x + 51/5

So I chose to find the slope of the other two points because parallel means that the m value is the same for both lines and only the y intercept will change. You can check this by plugging either point (-19,3) and (6,-2) into the y - y₁ = m ( x - x₁) equation and you will get the same answer. This is what I love about math, for fun you I hope can come up with -4/5 as the intercept of the line used to find the slope.

Please let me know if you have any other questions regarding this problem!

Answer 2

y-12 = 5(x+9)

y = 5x +57

the slope of the perpendicular is the negative inverse of -1/5, the slope of the line through the given two points (6,-2) and (-19,3). slope is rise over run or change in y values divided by change in x values, 3--2 over -19-6 = 5/-25 = -1/5

to get the negative inverse, change the sign and flip it upside down +5/1 = 5

plug in the point (-9,12) into

y-h = 5(x-k) where (k,h) = (-9,12) k= -9, h = 12

y-12 = 5(x+9)

y= 5x +45 + 12

y = 5x +57