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# Standard form review

Review linear standard form and how to use it to solve problems.

## What is the linear standard form?

This is the standard form of linear equations in two variables:
start color #11accd, a, end color #11accd, x, plus, start color #1fab54, b, end color #1fab54, y, equals, start color #e07d10, c, end color #e07d10
Usually in this form, start color #11accd, a, end color #11accd, start color #1fab54, b, end color #1fab54, and start color #e07d10, c, end color #e07d10 are all integers.

## Finding features and graph from standard equation

When we have a linear equation in standard form, we can find the x- and y-intercepts of the corresponding line. This also allows us to graph it.
Consider, for example, the equation start color #11accd, 2, end color #11accd, x, plus, start color #1fab54, 3, end color #1fab54, y, equals, start color #e07d10, 12, end color #e07d10. If we set x, equals, 0, we get the equation start color #1fab54, 3, end color #1fab54, y, equals, start color #e07d10, 12, end color #e07d10, and we can quickly tell that y, equals, 4, which means the y-intercept is left parenthesis, 0, comma, 4, right parenthesis.
In a similar way, we can set y, equals, 0 to get start color #11accd, 2, end color #11accd, x, equals, start color #e07d10, 12, end color #e07d10 and find that the x-intercept is left parenthesis, 6, comma, 0, right parenthesis. Now we can graph the line:
A first quadrant coordinate plane. The x- and y-axes each scale by one. The equation two x plus three y equals twelve is graphed. The points zero, four and six, zero are plotted.
Problem 1
What is the x-intercept of the line 5, x, minus, 2, y, equals, 10?
left parenthesis
comma, 0, right parenthesis
What is the y-intercept of the line?
left parenthesis, 0, comma
right parenthesis

Want to try more problems like this? Check out this exercise.

## Converting to standard form

In some cases (for example when solving systems of equations), we might want to bring an equation written in another form to standard form.
Let's bring the equation y, equals, start fraction, 3, divided by, 8, end fraction, x, plus, 5 to standard form:
\begin{aligned} y&=\dfrac{3}{8}x+5 \\\\ -\dfrac{3}{8}x+y&=5\quad\gray{\text{Put all variables on one side}} \\\\ -3x+8y&=40\quad\gray{\text{Multiply by denominator}} \end{aligned}
Problem 1
What is y, equals, minus, 2, x, minus, start fraction, 2, divided by, 7, end fraction in standard form?

Want to try more problems like this? Check out this exercise.

## Want to join the conversation?

• Doesn't the A in Standard form need to be positive?
• It depends, often definitions care more about the equation arrangement than the value of it, where A,B and C can be any real number. So, as long as you write it in Ax + By = C, it can be called standard form.

For the safe side, your teacher probably wants the A to be in positive integer value.

As an extra thought, to think about it, you can make an equation of line in many way (infinite way, actually)

Say 2x + 3y = 5
You can make it as:
4x + 6y = 10
-2x -3y = -5
16x + 24y = 40
2x/3 + y = 5/3
and many more!

If you graph this line, all of this create the same line (same slope and x,y intercept), only in different form of equation.
• 3x-y = -7 or -3x+y = 7 which is correct in standard form?
• Both are acceptable.
But, some textbooks differ on this subject and specify that the lead term must be positive. So, you should ask your teacher or check your textbook to make sure you pick the right one based on what is expected for your class.
• So- what do a, b, and c represent? I'm still confused on that.
• a, b, and c are variables that are "known", so when you are given an equation, you have specific values for a, b, and c. He gives several examples for specific values. x and y continue to variables that are "unknown." So this is the general form of the standard equation of a linear function, Sal notes that they should also be integers.
• I have a question, in the first graph, it shows 2x + 3y = 12.
Isn't it the other way around?
I tried to see the 2x + 3y in the graph, but it doesn't fit.
Correct me if i'm wrong, but it should be 3x + 2y = 12.

Thanks,
-Math4matt
• How do i write the equation of a line in standard form when i am given a word problem?
• It depends upon what info the problem gives you. You need to read it carefully.
Did it give you what looks like 2 ordered pairs? If yes, then you would:
1) Find the slope using the x & y values from the ordered pairs.
2) Use either slope-intercept form or point-slope form to get your initial equation.
3) Convert your equation to standard form.

Or, did the problem give you a slope (a rate of change) and 1 ordered pair? If this is the case, then you can just do steps 2 and 3 above.

Hope this helps.
• Why is it that we can "get rid of" the fraction by multiplying the other terms by its denominator? I guess beyond even that, why is it that when we multiply other terms by the denominator, it "goes away," but the numerator remains as an integer? AND, why is it that the sign of the fraction does not seem to have an impact? For example: y= -1/3x - 9 --> (multiply y and -9 by 3, the denominator,) --> 3y= -x -27
Why do we not multiply by negative 3? The fraction is a negative, so it seems like we should.
• Most commonly, we keep the negative with the numerator rather than saying the denominator is negative. It's just easier.

You could multiply by negative 3, but you end up making sign changes, and often can lead to errors.

We are allowed to multiply the equation by the denominator because the properties of equality let use multiply the equation by any value as long as we do the entire equation. The result is an equivalent equation to the original one.

Hope this helps.
• What is y={-\dfrac{1}{3}}x-9y=−
3
1

x−9y, equals, minus, start fraction, 1, divided by, 3, end fraction, x, minus, 9 written in standard form?
• I have been told that Ax + By +C = 0 is proper standard form. Is that correct?
• Ax + By + C = 0 can not be correct because it is not equivalent to standard from, ax + by = c.

When written as you suggest, it would have to be ax + by - c = 0. (You would have to subtract "c" from both sides to get there).
• what is 56=81x=98
• You have too many equals. Clarify your equation.
• How do I go from point slope to standard form
• Let's say your point slope equation is:
y - 3 = 2(x - 1)

P: Parentheses... you can't simplfiy what's inside the parentheses so move on to...

E: Expontents... you don't have any exponents so move on to...

MD: You have one multiplication problem 2(x-1) so use the distributive property of multiplication and get 2x - 2