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### Course: Algebra 1 > Unit 5

Lesson 5: Standard form- Intro to linear equation standard form
- Graphing a linear equation: 5x+2y=20
- Clarifying standard form rules
- Graph from linear standard form
- Converting from slope-intercept to standard form
- Convert linear equations to standard form
- Standard form review

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# Converting from slope-intercept to standard form

We can rewrite an equation in slope-intercept form (y=mx+b) to be in standard form (Ax+By=C) instead. In this example, we rewrite the slope-intercept equation y=2/3x+4/7 in standard form.

## Want to join the conversation?

- In standard form, shouldn't A be non-negative, non-zero?(43 votes)
- Incidentally, there is no restriction on A being zero. Indeed, B and C can also be zero (but not all at the same time). If A is zero then you get the equation By = C, or y = C/B, which is the equation of a straight, horizontal line with y-intercept equal to C/B. If B is zero then you get a straight, vertical line with x-intercept x = C/A, and if C is zero then you get the equation Ax + By = 0 which is a line with gradient -A/B passing through the origin.(5 votes)

- what if your slope intercept form numbers don't have fractions(22 votes)
- It's okay if they don't have fractions. It's still a linear equation.(16 votes)

- What does the m and b stand for in the equation y=mx+b?(13 votes)
`y=mx+b`

is the equation of a line`m`

is the slope of the line,

see: https://www.khanacademy.org/math/algebra-basics/alg-basics-graphing-lines-and-slope/alg-basics-slope/v/introduction-to-slope`b`

is the intercept, that is, the point where the line crosses the`y`

axis,

see: https://www.khanacademy.org/math/algebra-home/alg-linear-eq-func/alg-slope-intercept-form/v/slope-intercept-form(18 votes)

- Sal distributed the numbers at2:32to get rid of the fractions. Do you have to do that, or is that optional?(9 votes)
- If you are asked to write the equation in standard form, then you need to get rid of the fractions. Standard form is: Ax + By = C where A, B and C are integers (so no fractions).(15 votes)

- Why is converting the slope-intercept form into a standard form useful?(6 votes)
- Standard form has uses in matrices as well as solving for intercepts, although the y intercept is part of slope intercept form and x-intercept is not too hard to find. If you have a system of equations with one in slope intercept and one in standard, you could convert so that you could do matrices. Matrices help not as much with two variable equations because substitution or elimination are fine, but you need to learn matrices in two variables so that you may be able to do them in 3 or higher variable problems. Maybe there is no great reason for doing it, but it does help your math skills and may prepare you for higher math in the future.(15 votes)

- how do i answer 1/3x+y=8(9 votes)
- At about3:10, Sal multiplies part of the fraction by 21 and divides another part by 21. What is he doing and why?(9 votes)
- my teacher requires us to have 'Ax' be positive. Can anyone help me convert Sal's equation accordingly??(5 votes)
- If the Ax term is negative, you can multiply (or divide) the entire equation by -1. For example:

-1 (-14x+21y) = -1(12)

14x-21y = -12

Hope this helps.(5 votes)

- if they don't have fractions. It's still a linear equation.(7 votes)
- How would I go about converting a point-slope form equation (i.e
`y + 4 = -6(x + 6)`

) to standard form?(5 votes)

## Video transcript

- Rewrite Y is equal to 2/3x plus 4/7 in standard form. The equation must be simplified, which means all numbers must be integers that do not share a common
factor other than one. Alright, we'll worry about this
second part in a little bit. Let's see if we can rewrite this. So it's Y is equal to 2/3x plus 4/7. Let me get my scratch pad out. So it's Y is equal to 2/3x plus 4/7. So the way that it's written right now, this is slope intercept form. It's written in the form Y is equal to mx plus b, where m in this case is 2/3 and b is 4/7. It's very easy to figure
out what the slope and what the Y intercept
is from this equation. But we wanna write this in standard form. Which would be the form Ax plus By is equal to C. And that extra text they were saying, the equation must be simplified. Which means all numbers must be integers that do not share a common
factor other than one. That just means that A, B,
and C in the standard form they want need to be integers. And they want them to not
have any common factors. So if we got to the point of say 4x plus 2y is equal to 10, well this number, and this
number, and this number are all divisible by two. They all have the common factor of two. So we would want to simplify it more. Divide them all by two
and then you would get two so you divide this by two. You get 2x, divide this by two you'll get plus y is equal to 5. So this is the form
that they're asking for and probably because it's just easier for the site to know that
this is the right answer. Because there's obviously a
bunch of forms in this way. So let's see if we can do that. Let's see if we can write
it in standard form. So the first thing I would wanna do is, well there's a bunch of ways
that you could approach it. The first thing you could try to do is well let's get rid of
all of these fractions. And the best way to get
rid of the fractions is to multiple by three
and to multiply by seven. If you multiply by three you
get rid of this fraction. If you multiply by seven you
get rid of this fraction. So if you multiply three
and you multiply by seven. Let me just rewrite it over here. There's actually also a couple
of ways that we can do this. So if you multiply. So one way to do it. So we start with Y is equal to 2/3x plus 4/7. So if I multiply this side by
three and I multiply by seven, I have to do that to this side as well. So this is going to be multiplied by three and multiplied by seven. So the left hand side becomes 21y. 21y. Three times seven of course is
21, we just figured that out. We would distribute the 21. 21 times 2/3, well let's see. 21 divided by three is
seven, times two is 14. So it's gonna be 14x. And then 21 divided by seven
is three times four is 12. So just like that I was able
to get rid of the fractions. And now I wanna get all the X's and Y's on one side. So I wanna get this
14x onto the left side. So let's see if I can do that. So I'm gonna do that by. To get rid of this I would
want to subtract 14x. I can't just do it on the right hand side I have to do it on the
left hand side as well. So I wanna subtract 14x, and
then what am I left with? Let me give myself a
little bit more space. So on the left hand side I have negative 14x plus 21y. Plus 21y is equal to. Let's see and I subtracted
14x to get rid of this. And then I have this is equal to 12. Now let's see, am I done? Do these share any, do 14, 21, and 12 share any common factors? Let's see, 14 is divisible
by two and seven. 21 is divisible by three and seven. 12 is divisible by two, six, three, four. But all of these aren't
divisible by the same number. 14, let's see. 14 is divisible two,
so is 12, but 21 isn't. 14 is divisible by seven,
so is 21, but 12 isn't. And 21 and 12 are divisible
by three but 14 isn't. So I think this is about as
simplified as I could get. If there was a common factor
for all three of these numbers then I would divide all
of them the way I did in that previous example. But that's not the case right over here. So it's negative 14x
plus 21y is equal to 12. So let me see if I can
remember that and type that in. So it is, negative 14x plus 21y is equal to 12. Now let's see if I got it right. It worked out.