Question

determine whether (4,-3) and (0,6) are solutions of 2x+y=5

Answer 1

Plug and play.

Answer 2

You want to find the equation for a line that passes through the two points:
(4,-3) and (0,6).
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:
\[m=\frac{ y_2-y_1 }{ x_2-x_1 }\]
So what we need now are the two points you gave that the line passes through. Let's call the first point you gave, (4,-3), point #1, so the x and y numbers given will be called x1 and y1. Or, x1=4 and y1=-3.

Also, let's call the second point you gave, (0,6), point #2, so the x and y numbers here will be called x2 and y2. Or, x2=0 and y2=6.

Now, just plug the numbers into the formula for m above, like this:
\[m=\frac{ 6--3 }{ 0-4 }\]
m= - 9/4

So, we have the first piece to finding the equation of this line, and we can fill it into y=mx+b like this:
\[y=-\frac{ 9 }{ 4 }x+b\]

Now, what about b, the y-intercept?
To find b, think about what your (x,y) points mean:
(4,-3). When x of the line is 4, y of the line must be -3.
(0,6). When x of the line is 0, y of the line must be 6.

Because you said the line passes through each one of these two points, right?

Now, look at our line's equation so far: y=-9/4x+b.
b is what we want, the -9/4 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 (4,-3) and (0,6).

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:

(4,-3). y=mx+b or -3=-9/4 × 4+b, or solving for b: b=-3-(-9/4)(4). b=6.
(0,6). y=mx+b or 6=-9/4 × 0+b, or solving for b: b=6-(-9/4)(0). 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
(4,-3) and (0,6)

is

y=-9/4x+6

Answer 3

@princessmommy