A line goes through the points (3,9) and (1,1) write an equation of this line in standard form

Question

Geometry question that i don't quite understand Because I don't have understand how to get from standard form wirh just points. 

Answer 1

Hi Amanda,

 

There are a few different ways to express the equation of a line:

 

    The slope-intercept format: y = mx + b       where "m" is slope and "b" is the y-intercept

 

    The point-slope format: (y-y₁) = m(x-x₁)     where "m" is slope, and (x₁,y₁) is a point on the line

 

    And the standard form:  Ax + By = C           where A,B, and C are all integers, and A should be positive

 

If we can figure out ONE of the forms of the equation of a given line, we can use algebra to turn it into either of the other two. If we are given at least one of the points, then the point-slope format is usually the easiest to find. We already know TWO points that the line goes through, so we could pick either of them as our (x₁,y₁). But, we do still need to find the slope.

 

    Slope = rise/run = (change in y)/(change in x) = (9-1)/(3-1) = 8/2 = 4

 

So, a point-slope representation for this line would be (and I'll use the (1,1) point for (x₁,y₁)):

 

    (y-1) = 4 (x-1)

 

Now we just need to use algebra to rearrange it into the Ax + By = C format:

 

First I'll distribute the 4 on the right hand side of the equation:

 

    y - 1 = 4x - 4

 

The number in front of the x is already positive, so I will leave it there, and move the "y" term to the right side, and the constant numbers to the left.

 

I'll subtract y from both sides:

 

    -1 = 4x - y - 4

 

And now I will add 4 to both sides:

 

    3 = 4x - y

 

So the standard form of this line is:

 

    4x - y = 3

 

I hope this helps!

 

Andy

Answer 2

Standard form is Ax + By = C.

 

Usually the easiest way to get to standard form is by first finding the "slope intercept" form of the equation and then converting it to standard form.  This is y = mx + b, where m is the slope and b is the y-intercept, the value of y where the line crosses the y-axis.

 

Step 1:  Find the slope.  This is the change in y between the two points divided by the change in x, or (y₂ - y₁) / (x₂ - x₁).  In this case, that's (9-1) / (3 -1) = 8/2 = 4.

 

Step 2:  Find the y-intercept.  Substitute x and y from one of the points, and m, into the equation to find b.  So

 

1 = (4)(1) + b

1 = 4 + b

b = -3

 

So the slope intercept form is y = 4x - 3.

 

To get the standard form, just rearrange the terms so that all the variables are on the left side and keep the constant on the right.  It is best to put x first on the left side when you do this:

 

y = 4x - 3

-4x + y = -3

 

And that solves the problem!  A = -4, B = 1, and C = -3.