Answer:
1. y=2x+6 or 2x - y = -6
2. x = -1, y = 7
3. y = −3x/2 − 3/2
Step-by-step explanation:
For the first picture:
You want to find the equation for a line that passes through the two points:
(-2,2) and (1,8).
First of all, remember what the equation of a line is:
y = mx+b
Where:
m is the slope, and
b is the y-intercept
First, let's find what m is, the slope of the line...
The slope of a line is a measure of how fast the line "goes up" or "goes down". A large slope means the line goes up or down really fast (a very steep line). Small slopes means the line isn't very steep. A slope of zero means the line has no steepness at all; it is perfectly horizontal.
For lines like these, the slope is always defined as "the change in y over the change in x" or, in equation form: m = y2 - y1 / x2 - x1
So what we need now are the two points you gave that the line passes through. Let's call the first point you gave, (-2,2), point #1, so the x and y numbers given will be called x1 and y1. Or, x1=-2 and y1=2.
Also, let's call the second point you gave, (1,8), point #2, so the x and y numbers here will be called x2 and y2. Or, x2=1 and y2=8.
Now, just plug the numbers into the formula for m above, like this:
m = 8 - 2/1 - -2
or m = 6/3
or m = 2
So, we have the first piece to finding the equation of this line, and we can fill it into y=mx+b like this:
y=2x+b
Now, what about b, the y-intercept?
To find b, think about what your (x,y) points mean:
(-2,2). When x of the line is -2, y of the line must be 2.
(1,8). When x of the line is 1, y of the line must be 8.
Because you said the line passes through each one of these two points, right?
Now, look at our line's equation so far: y=2x+b. b is what we want, the 2 is already set and x and y are just two "free variables" sitting there. We can plug anything we want in for x and y here, but we want the equation for the line that specfically passes through the two points (-2,2) and (1,8).
So, why not plug in for x and y from one of our (x,y) points that we know the line passes through? This will allow us to solve for b for the particular line that passes through the two points you gave!.
You can use either (x,y) point you want..the answer will be the same:
(-2,2). y=mx+b or 2=2 × -2+b, or solving for b: b=2-(2)(-2). b=6.
(1,8). y=mx+b or 8=2 × 1+b, or solving for b: b=8-(2)(1). b=6.
See! In both cases we got the same value for b. And this completes our problem.
The equation of the line that passes through the points
(-2,2) and (1,8)
is
y=2x+6
For the second picture:
8x+3y=13;3x+2y=11
Multiply the first equation by 2,and multiply the second equation by -3.
2(8x+3y=13)
−3(3x+2y=11)
Becomes:
16x+6y=26
−9x−6y=−33
Add these equations to eliminate y:
7x=−7
Then solve7x=−7for x:
7x=−7
7x/7 = -7/7 (Divide both sides by 7)
x = -1
Now that we've found x let's plug it back in to solve for y.
Write down an original equation:
8x+3y=13
Substitute−1forxin8x+3y=13:
(8)(−1)+3y=13
3y−8=13(Simplify both sides of the equation)
3y−8+8=13+8(Add 8 to both sides)
3y=21
3y/3 = 21/3 (Divide both sides by 3)
y = 7
For the third picture:
The equation of the line in the slope-intercept form is y=2x/3 + 1
The perpendicular slope (
m1) is negative reciprocal of the slope m
m1 = -1/m = -1 divide by 2/3 = -3/2
The slope of the perpendicular line is negative inverse: m = −3/2
So, the equation of the perpendicular line is y=−3x/2 + a
To find a, we use the fact that the line should pass through the given point: −6 = (−3/2) ⋅ (3) + a
Thus, a = −3/2
Therefore, the equation of the line is y = −3x/2 − 3/2
