whats the equation for line a passes through (-2,-5) and (0,-1); line b passes through (3,1)and (1,-3)

Question

i need help on this because its hard and i dont understant it

Answer 1

Hi Michelle,

 

The most important aspect is to note that we are searching for an equation in the form, y = mx + b. In order to get to that, we can expand this equation out into what is called the point-slope formula, y - y1 = m(x -x1), where m is the slope of any two points on the line and x1 and y1 are any two points on the line. 

 

Let's find the slope, m,  first. We can work with the two given points (-2, -5) and (0, -1). Remember, the number left of the comma or first in each point is on the x-axis or the abscissa. That's -2 and 0. The numbers to the right of the comma in each point are on the y-axis or the ordinate. Those numbers are -5 and -1, from our example. Using that knowledge, we can now find the slope...

 

m = Δy / Δx

 

Let Δy = the difference in our numbers for the y-axis, which is -1 - (-5). 

 

-1 - (-5) = -1 + 5 = 4.  So, Δy = 4.

 

Let's look at Δx, our difference is the numbers on the x-axis. Using -2 and 0 from our points, we have 0 - (-2).

 

0 - (-2) = 0 + 2 = 2. So Δx =2.

 

Applying our above formula, m = Δy / Δx = 4 /2 = 2. So, our slope is 2 or m =2.

 

Note that I selected the numbers in the second point for each calculation. We could have easily selected the numbers from the first point and achieved the same result! 

 

Referring back to our point-slope formula, we insert 2 in for m and we now have

 

                y - y1 = 2(x -x1)

 

We can now select either point (-2, -5) or (0, -1) for x1 and y1. Let's select (0, -1). I always try and select points that might have a zero in them (the calculation is oftentimes quicker!).

 

Selecting (0, -1), let x1 = 0, since 0 is the x-coordinate. And let y1 = -1, since -1 is the y-coordinate. 

 

Inserting them into our point-slope formula gives us:

 

                y - (-1) = 2(x - 0)

 

Reducing our equation, we get:

 

                 y +1 = 2x + 0,

 

which is,

 

                 y + 1 = 2x.

 

Solving for y, we add (-1) to both sides of the equation

 

                 y + 1 +(-1) = 2x -1,

 

we have

 

                 y = 2x -1.    

 

That is our answer.

 

VERIFICATION or checking our solution:

If you want to verify, insert the x-coordinates into the equation and see if your answer for y corresponds to each of the numbers for y in each of your given points (-2, -5) and (0, -1).

 

Let's have a look:

Inserting (-2) into y = 2x - 1,

 

                 y = 2(-2) - 1 = (-4) -1 = -5.

 

               -5 is the y-coordinate in our first given point, (-2, -5), which checks out.

 

Let's look at the second point (0, -1). Insert the x-coordinate, 0, into our equation, y = 2x -1.

 

               y = 2*0 -1 = 0 - 1 = -1.

 

               -1 is the y-coordinate of the second point, (0. -1).  Therefore, since both of the y-coordinates can be found by inserting each of their x-coordinates into our equation, our solution, y =2x -1 is correct!

 

Now try this for (3, 1) and (1,-3).  :-)

 

 

 

Answer 2

For any line y= mx + b where m is the slope. You have two points, so you can compute the slope as delta y/delta x and delta y = (-1--5) = (-1 + 5) = 4 and delta x = (0- -2) = 2. So slope equals 4/2 = 2. Thus, your equation now looks like y = 2x + b, you need to figure out what b is. That's easy. Plug in the values of any given point and solve for b. I'll pick (0, -1), so -1 = 2 * (0) + b. Therefore, b = -1 and the equation becomes y = 2x -1. You can check this by seeing if the coordinates of the other point work. Does -5 = 2*(-2) - 1? Yes. So, you are good. Do the same thing with line b.