How do I write the equation of a line?

Question

Sorry for all the math questions. I am just a math idiot.

1. Slope: 2/3, y-intercept: -4

2. Passes through points (1,2) and (-4,7)

3. Pases through (-2,3) and is perpendicular to the line y = -2x +5

4. Passes through (4,6) and is parallel to the line that passes through (4,-6) and (8,-4)

Sorry for all the questions. If you can answer any I would appreciate it :]

Answer 1

The format for a line equation is y=mx+b where m is the slope and b is the y-intercept.

1. Slope: 2/3, y-intercept: -4

This one is pretty easy because the question gives you what you need:

y= m  x + b

y=2/3 x -  4

2. Passes through points (1,2) and (-4,7)

Get your slope first.  You get your slope by getting the change in y over change in x, or

m = (y1 - y2) / (x1 - x2) which is

m = (2  -  7 ) / (1 - (-4)) simplified to

m =     -5      /    5  

m = -1.  So -1 is your slope.

With the slope and any one point we can derive the y-intercept using the same formula.  Except this time we will use one known point (1,2), the slope m=-1, and (0,b) as the y-intercept.  We use 0 for x on this y-intercept because any y-intercept would have to have a 0 value for x.

m = (y1 - y2) / (x1 - x2) which is

-1 = (2   - b) / (1  -  0) simplified is

-1 = 2 - b which is

-3 = -b which is

b = 3

So, with slope of m=-1 and y-intercept of b=3 we can go from y=mx+b to:

y = -1x +3  

3. Pases through (-2,3) and is perpendicular to the line y = -2x +5

To solve this, you need to know that when you have two perpendicular lines, their slopes are related to each other by this formula:

m= 1 / -m1

So.. y = -2x +5 means that the slope of this line is -2, therefore the slope of YOUR line is

m = 1/-(-2) or m = 1/2  

So you have your slope.  Next you need the y-intercept.  Same as in Problem 2:

m = (y1 - y2) / (x1 - x2) which is

1/2 = (3   - b) / (-2  -  0) simplified is

1/2  = -3/2 + b/2 times 2 on both sides is

1     = -3 + b

b = 4

y=mx+b becomes

y=1/2x + 4

4. Passes through (4,6) and is parallel to the line that passes through (4,-6) and (8,-4)

This one is actually easier than #3 above.  Parallel lines have the same slope, so you get your slope from the live formed by (4,-6) and (8,-4)

m = (y1 - y2) / (x1 - x2) which is

m = (-6 - (-4))/ (4   -  8)

m = -2/4

m= 1/2

Now you need the y-intercept (0,b) from this formula:

m = (y1 - y2) / (x1 - x2) which is

1/2 = (6   - b) / (4  -  0) simplified is

1/2 = 6/4 - b/4  times 4 on both sides is

2 = 6 - b

b=4

y=mx+b becomes

y=1/2x + 4 (which is suspicious because it's the same exact lilne as in problem #3)

I hope some of this helps.  Remember if you want to double check your answers, just graph the line you come up with and see if it satisfies the question (what points it passes through, etc)

Good luck

Answer 2

The equation of a line is also known as Slope-Intercept form.

Slope-Intercept form in terms of math is:

y=mx+b

m is the slope, and b is the y-intercept. The y-intercept is where your line crosses, or "intercepts" the y-axis of your graph.

The formula for slope is:

rise/run (this is basically rise divided by run)

To solve question 1, you need to just plug in the numbers given into the m and b variables, as seen here:

y = (2/3)x + (-4)

The final result will be:

y = 2/3x - 4

For question 2, you remember that slope is rise/run, right?

To find the slope with two given coordinates, you use the formula:

(Y2 - Y1)/(X2 - X1)

Note that the 2's and 1's are just the first and second Y's and X's, not real numbers.

To find the second Y, look for the Y in the second set of coordinates. In this case, it's 7. To find the first Y, look for the Y value in the first set of coordinates. In this case, it's 2. Use this same method to find the X's, and you get:

(7 - 2)/[-4 - 1]

You will arrive at -5/5, which is equal to -1, but we're not done yet.

You can write the equation out as:

y = -1x + b

but now we have to find b. You find b by taking any of the two set of coordinates (preferrably the easier one) and plugging in the x and y values:

2 = -1(1) + b

b = 3

So the full equation is:

y = -x + 3 (you can just drop the 1)

For question 3, you have to remember that perpendicular lines mean that the slopes are a reciprocal (the opposite) of each other. To solve problem 3, you have to reciprocate -2 and change it into 1/2. Then you have to plug in the values for X and Y:

(3) = 1/2(-2) + b

b = 4

The final answer is:

y = 1/2x + 4

For question 4, apply what you've learned in the previous problems, except the only difference is that the line is parallel, meaning the slopes do not have to be reciprocated; they stay the same.

First, you have to find the slope:

m = [-4 - (-6)]/(8 - 4)

m = 2/4 = 1/2

Now you have to plug in the values for X and Y to find B.

y = mx + b

(6) = 1/2(4) + b

6 = 2 + b

b = 4

So the final answer for question 4 is:

y = 1/2x + 4

Answer 3

<< Slope: 2/3, y-intercept: -4 >>

The standard format of the equation of a straight line is

y = mx  + b

where

m = slope

b = y-intercept

So, the equation for the given data that you have is

y = (2/3)x - 4

<< Passes through points (1,2) and (-4,7) >>

By definition,

m = slope = delta y/delta x

m = (7 - 2)/(- 4 - 1) = 5/(-3) = -5/3

Take any of the two given points -- take (1,2) -- and the equation of thie line is

-5/3 = (y - 2)/(x - 1)

Simplifying,

-5(x - 1) = 3(y - 2)

-5x + 5 = 3y - 6

Rearranging,

3y = -5x + 11 --- this is the equation of the straight line that will satisfy the given data in the problem.

If you want to reduce the above equation to conform with the standard straight line equation, then

y =-(5/3)x + 11/3

<< Pases through (-2,3) and is perpendicular to the line y = -2x +5 >>

The slope of the given line is m = -2. And if another line is perpendicular to this given line, then its slope is 1/2.

Thus being said,

1/2 = (y - 3)/(x - -2)

1/2 = (y - 3)/(x + 2)

(x + 2) = 2(y - 3)

x + 2 = 2y - 6

2y = x + 8

y = (1/2)x + 4

<<. Passes through (4,6) and is parallel to the line that passes through (4,-6) and (8,-4) >>

Slope of line that passes through (4, -6) and (8, -4) is

m = (-4 - -6)/(8 - 4) = 2/4 = 1/2

Since the line that passes through (4,6) is parallel to the other line, then the slopes of both lines are equal (m = 1/2).

Therefore,

1/2 = (y - 6)/(x - 4)

x - 4 = 2(y - 6)

x - 4 = 2y - 12

2y = x + 8

y = (1/2)x + 4

Hope this helps you understand the concept of slopes. Good luck.

Answer 4

Y=mx +b

Slope is the change in y / change in x (y2 -y1) / (x2-x1)

perpendicular.. slope is inverse.....EX (line with slope of 2, a line perpendicular has a slope of 1/2

you can do these

Answer 5

First - memorize y=mx+b if you can, it's used a lot.  m is the slope, b is the y-intercept, and y and x are coordinates.

1. m = 2/3, b = -4

So y = (2/3)x - 4

2. Use point-slope form to solve this one, which is m = (y1 - y2) / (x1 - x2).

m = (2-7) / (1-(-4))

m = 5 / 5

m = 1

Now that you have the slope, you can use it and one of the points given to solve it.  Use slope-intercept for this (I use the point (-2,3)) -

y = mx + b

3 = 1 (-2) + b

3 = -2 + b

5 = b

So now that you have m and b, you can create:

y = x + 5

(mx would just be 1x, so I didn't add the 1)

3.  You know it is perpendicular to y = -2x + 5, which is y = mx + b - so it's perpendicular to a line with a slope of -2.  

Perpendicular lines have slopes that are inverse reciprocals of the original - so "flip" -2, and make it positive - 1/2.  The slope, m, of the line you're looking for is 1/2.

Now, similar to 2, plug what you know into y=mx+b.

y = mx + b

3 = (1/2)(-2) + b

3 = -1 + b

4 = b

Now you know m = 1/2 and b = 4, so...

y = (1/2)x + 4

4.  First find the slope of the line it is parallel to, just like in 2.

m = (-6 - (-4)) / (4 - 8)

m = -2 / -4

m = 1 / 2

Now that you know this, plug the info you know into y=mx+b:

y = mx + b

6 = (1/2)(4) + b

6 = 2 + b

4 = b

Now, put m and b into y=mx+b -

y = (1/2)x + 4

Hope that's what you needed.

...looks like someone beat me to answering it, lol.