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- Lilmike234

What is the equation, in standard form, of the line passing through the points (2,3) and (4,2) ?

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- Lilmike234

What is the equation, in standard form, of the line passing through the points (2,3) and (4,2) ?

- schrodinger

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- mathstudent55

First, find the slope of the line.
You can use the two given points for that.

- mathstudent55

The slope of the line that passes through points \((x_1, y_1)\) and \((x_2, y_2) \) is
\(slope = m = \dfrac{y_2 - y_1}{x_2 - x_1} \)

- mathstudent55

To find the slope, subtract the y-coordinates, and divide by the difference of the x-coordinates. Make sure you subtract the y-coordinates and the x-coordinates in the same order.
Can you try this and tel me what you get?

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- Lilmike234

-1/2 ?

- mathstudent55

Correct.
|dw:1437619975077:dw|

- mathstudent55

Notice what I did above. I did it both ways. Using one point first one time and the other point first the other time to do the subtractions. It does not matter which point you choose as the first point. The important thing is to not mix the order when you subtract. As you can see both ways give the same answer, and you got it right.

- mathstudent55

Now that we have the slope, we do the next step.

- mathstudent55

The equation of a line given two points is:
\(y - y_1 = \dfrac{y_2 - y_1}{x_2 - x_1} (x - x_1)\)
or simply
\(y - y_1 = m (x - x_1)\) when you let m = the slope.
y and x above are the y and x of the equation.
\(x_1\) and \(y_1\) are the x- and y-coordinates, respectively, of a point on the line.

- mathstudent55

We already have the slope, so we can use the form
\(y - y_1 = m(x - x_1)\)
Since we know the \(slope = m = -\dfrac{1}{2} \), we replace m with the slope
\(y - y_1 = -\dfrac{1}{2} (x - x_1) \)

- mathstudent55

All that is left to do is to replace \(x_1\) and \(y_1\) with the coordinates of one of the points. You can use whichever point you want of the two given points.

- mathstudent55

Let's use point (2, 3). We replace \(x_1\) with 2 and \(y_1\) with 3.
\(y - 3 = -\dfrac{1}{2}(x - 2)\)

- mathstudent55

You already have the equation of the line.
Now you can put the equation in standard form or slope-intercept form if you'd like to.

- Lilmike234

y=5/2x+8

- Lilmike234

?

- mathstudent55

\(y - 3 = -\dfrac{1}{2}(x - 2)\)
\(2y - 6 = -(x - 2) \)
\(2y - 6 = -x + 2\)
\(2y = -x + 8\)
\(y = -\dfrac{1}{2} x + 4\)

- Lilmike234

The answer choices are
1. y=5/2x+8
2. 5x+2y=16
3. 2x-3y=9
4. 5x-2y=16

- mathstudent55

Are you sure those are the choices of the same problem?

- Lilmike234

Yes

- mathstudent55

Can't be.

- mathstudent55

We have a slope of -1/2
None of those choices have a slope of -1/2

- Lilmike234

Well those were the answer choices

- mathstudent55

Let's look at your choices and put every equation in the slope intercept form,
y = mx + b.
1. y=5/2x+8 slope = 5/8
2. 5x+2y=16
2y = -5x + 16
y = -5/2 x + 8 slope -5/8
3. 2x-3y=9
-3y = -2x + 9
y = 2/3 x - 3 slope 2/3
4. 5x-2y=16
-2y = -5x + 16
y = 5/2x - 8 slope 5/2
As you can see, none of your choices have slope -1/2
None of them are answer to this question.
Are you sure these are the given answers to this question and are you sure you copied the points correctly when you posted the question?

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