Write an equation of the line that passes through the points 2 5 0 5

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).  :-)

2 Answers By Expert Tutors

Hello, Crisherly,

The equation format for a straight line is y = mx + b, where m is the slope and b is the y-intercept, the value of y at x = 0. The slope is the rate of change of the line as x changes values. It is also termed the "rise over the run," of the difference in the value of y as x is changed.

We can calculate the rise/run from the two points. (0,2) and (-4,5). The rise is the difference between the two y values (5 - 2 = 3) and the run the difference in the x values (-4 - 0 = -4). Rise/run is therefore -3/4. This is "m" in the equation.

y = -(3/4)x + b

We need to find b. Input one of the points, (0,2) and solve for b.

2 = -(3/4)*0 + b

b = 2 (the line crosses the y axis when x = 0).

The full equation becomes:

y = -(3/4)x + 2

Bob

Karina F. answered • 12/10/20

If you seek success...I am here to help

When given TWO points on a line; you will need to calculate the slope, m and use the point-slope formula to find the equation of the line

To find the slope, m = rise/run, it does NOT matter which point is P1 and P2, the slope will be the same.

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

I use will P1 = (0, 2) and P2 = (-4, 5)

m = (5 - 2) / (-4 - 0) = -3/4

Now use the point-slope formula and use just ONE of the points given and the slope:

(y - y1) = m(x - x1)

I will use P1 = (0, 2)

y - 2 = -3/4(x - 0)

y - 2 = -3/4(x)

Isolate the y on the LHS by +2 on both sides

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

If you're curious try P2 as a check. P2 = (-4, 5)

(y - y1) = m(x - x1)

y - 5 = -3/4(x - (-4)) = -3/4(x + 4)

y - 5 = -3/4(x) -3/4(4)

The 4s cancel out on the RHS and you're left with -3

y - 5 = -3/4(x) - 3

Isolate the y on the RHS by +5 on both sides

y = -3/4(x) - 3 + 5

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

Hint: Always use #s that will make your calculations as easy as possible.

Hope this helps :)

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