Input in standard form the equation of the given line. The line that passes through (1,5) and (-2,3)

Let's break this down into a few parts first.

Slope intercept form is otherwise known as y = mx + b, where m is the slope and b is the y-intercept.
Parallel means that the two lines will never meet. They are parallel to each other. In math terms, their slopes are the same, so the m values must be the same.

Starting with y = -5x + 1, putting into slope intercept form, y = mx + b, m = -5 and b=1.

Since it is parallel, it must have the same slope and the m values are the same.

So, y = -5x + b, but we don't know what the value of b is. To determine this, we know the point (-4,-6) is on the line that we are trying to find, so we can substitute it into the equation and calculate b to make it work.

Every time we see an x we put in -4 and every time we see a y, we put in -6.

-6 = -5(-4) + b
-6 = 20 + b
subtract 20 from both sides
-6 - 20 = 20 + b - 20
-26 = b
Now substitute this into the equation.

y = -5x + -26
Putting it into correct form, we get y = -5x - 26.

Let's check it to see if it is correct.

It has a slope of -5, so it is parallel to y=-5x + 1

Is the point (-4,-6) on the line? Let's substitute it in to see if it is on the line.

Again, everywhere we see an x, we put in -4 and everywhere we see a y we put in -6.

-6 = -5(-4) - 26
-6 = 20 - 26
-6 = -6

Sorry for the very long explanation, but after you do a few of them, you will be able to knock them off in minutes.

Good luck,

Paul

Let's break this down into a few parts first.

Slope intercept form is otherwise known as y = mx + b, where m is the slope and b is the y-intercept.
Perpendicular means that the two lines will intersect each other at a 90 degree angle. In math terms, their slopes will be negative reciprocals of each other.
Starting with y + 4 = 1/4(x -7), putting into slope intercept form, y = mx + b, m = 1/4 and b=-7/4 -4.

Since it is perpendicular, it must have a slope that is a negative reciprocal. The slope of the original line is 1/4, so the multiplying it -1, we get -1/4, and taking the reciprocal, we get -4. So the slope of the perpendicular line is -4.

So, y = -4x + b, but we don't know what the value of b is. To determine this, we know the point (-4,7) is on the line that we are trying to find, so we can substitute it into the equation and calculate b to make it work.

Every time we see an x we put in -4 and every time we see a y, we put in 7.

y = -4x + b
7 = -4(-4) + b
7 = 16 + b
subtract 16 from both sides
7 - 16 = 16 + b - 16
-9 = b

Now substitute this into the equation.

y = -4x + -9
Putting it into correct form, we get y = -4x - 9.

Let's check it to see if it is correct.

It has a slope of -4, so it is perpendicular to y=1/4x - 5 3/4.
Is the point (-4,7) on the line? Let's substitute it in to see if it is on the line.

Again, everywhere we see an x, we put in -4 and everywhere we see a y we put in 7.

y= -4x - 9
7 = -4 (-4) - 9
7 = 16 - 9
7 = 7

Sorry for the very long explanation, I was being overly thorough, but after you do a few of them, you will be able to knock them off in minutes.

Good luck,

Paul

2- 4 - 8 - 16

A math teacher might know the answer to that one. Do you know a math teacher you could ask?

The "slope intercept form of the equation of a (the) line" is y=mx+b, where m is the slope of the line and b is the y-intercept.

We are given the slope of 1/2, so m= 1/2.

We can now write y=1/2 x + b.

Since the point (-2,-3) is on the line, we can substitute it in and solve for b. We put the -2 in for x and -3 in for y.

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

Thus, the equation of the line is y= 1/2 x -2

To check if we did this correctly, plug in the point (-2, -3) to see if it works.
Left Side Right Side
-3 = 1/2 (-2) -2
= -1-2
= -3

Assuming the 'standard form' is "slope-intercept", calculate the slope from the equation m = y2-y1 = 5 - 1 = 4 = -2
x2-x1 4 - 6 -2
The intercept can be found by substituting either of the two points into the equation y = mx + b
5 = (-2)4 + b
5 = (-8) + b
13 = b
(OR, using the other point, y = mx + b
1 = (-2)6 + b
1 = (-12) + b
13 = b )
Then expressing in general:
y = (-2) x + 13

Being parallel to the given line, the equation of the line you are seeking has the same slope, which in this case is a=1/4.
So the equation sought is as follows
y=(1/4)x +b, where b is to be found.
To find b, use the stated fact that the line passes through the point (x=8, y=-1). All that means is that the point (8,-1) is on the line whose equation you are looking for. If it is on the line with equation y=(1/4)x+b
then its coordinates x=8, and y=-1 satisfy the relation y=(1/4)x+b. In other words, if you substitute 8 for x, and -1 for y, the equality holds true -1=(1/4)*8 +b
This gives you a way to find the initial value of the function (the y-intercept b ). Just solve -1=(1/4)*8 +b to find b.
I leave this pleasure to you.

1. Transform this equation to its functional form:
2. 9y=-3x+17 or y=(-1/3)x+17/9
3. In the last equation, the slope is the coefficient of x, namely -1/3.
   
4. A line parallel to this one must have the same slope (-1/3).
5. So the equation of your line starts this way: y=(-1/3)x+b.
6. To identify (calculate) b, you must make use of the fact that the parallel line passes through the point (1,5).
   
7. That means that the coordinates of the point (1,5) satisfy the equation of the parallel line y=(-1/3)x+b
8. Substitute 5 for y, and 1 for x and solve for b.

And that is as far I will go. I leave it to you finish up the work : find b and write the equation in the functional form, then convert it to the general form (Ax+By+C=0) if that is what you are asked to produce.

Calcualte the slope of the line as
a=(7-(-11))/(10-(-5))=18/15=6/5
Use the fact that the line passes through one of the two points, for example (10,7)
7=(6/5)*10+b=12+b
Obtain b as b=7-12=-5
The equation of the line in functional form is y=(6/5)x-5
Multiply everything by 5 to clear the fraction
5y=6x-25 or 0=6x-5y-25
Finally, the equation in general form (standard?) is 6x-5y-25=0.

Check the calculation by verifying that the point (10,7) lies on the line.
6(10)-5(7)-25=60-35-25=60-60=0 CHECKed!
Check that the second point (-5,-11) lies on the line also (if you want to)
6*(-5)-5*(-11)-25=-30+55-25=0
That checks OK.

Calculate the slope (gradient) of the line as a=(y2-y1)/(x2-x1) where y2=1, y1=0, x2=0, and x1=-6. You should get a=(1-0)/(0-(-6))=1/6
The y-intercept is the y-cordinate for x=0. Its value is 1.
The equation is then y=(x/6) 1.