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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.
Want to learn more about standard form? Check out this video.

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
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
comma, 0, right parenthesis
What is the y-intercept of the line?
left parenthesis, 0, comma
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
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:
y=38x+538x+y=5Put all variables on one side3x+8y=40Multiply by denominator\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?
Choose 1 answer:
Choose 1 answer:

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

Want to join the conversation?

  • starky seedling style avatar for user Lo
    Doesn't the A in Standard form need to be positive?
    (24 votes)
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    • aqualine ultimate style avatar for user Apple
      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.
      (23 votes)
  • starky sapling style avatar for user Leilah McIntosh
    3x-y = -7 or -3x+y = 7 which is correct in standard form?
    (8 votes)
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    • stelly blue style avatar for user Kim Seidel
      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.
      (6 votes)
  • primosaur ultimate style avatar for user Elijah Merrill [Gil-Galad]
    So- what do a, b, and c represent? I'm still confused on that.
    (2 votes)
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    • mr pink green style avatar for user David Severin
      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.
      (9 votes)
  • hopper cool style avatar for user math4matt
    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
    (4 votes)
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  • blobby green style avatar for user chelsy92
    How do i write the equation of a line in standard form when i am given a word problem?
    (5 votes)
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    • stelly blue style avatar for user Kim Seidel
      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.
      (5 votes)
  • blobby green style avatar for user aliciaafinney
    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.
    (3 votes)
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    • stelly blue style avatar for user Kim Seidel
      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.
      (2 votes)
  • blobby green style avatar for user dw1589895
    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?
    (3 votes)
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  • aqualine ultimate style avatar for user bradizzad
    I have been told that Ax + By +C = 0 is proper standard form. Is that correct?
    (2 votes)
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  • mr pink red style avatar for user Aryan Mantri
    what is 56=81x=98
    (2 votes)
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  • blobby green style avatar for user zberman21
    How do I go from point slope to standard form
    (0 votes)
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    • leafers ultimate style avatar for user KathyC
      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
      Now your equation is
      y - 3 = 2x - 2

      AS: Addition and Subtraction... combine the similar terms. The only ones you have are the constants:
      y - 3 + 3 = 2x - 2 + 3
      y = 2x + 1 (This is the slope-intercept form of the equation.)

      Move the "x's" to the right hand side.
      y - 2x = 1

      Rearrange so the x term is in front
      -2x + y = 1 (This is the standard form of the equation)
      (8 votes)