How to Convert Slope Intercept Form to Standard Form

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Any equation that relates the first power of x to the first power of y produces a straight line on an x-y graph. The standard form of such an equation is Ax + By + C = 0 or Ax + By = C. When you rearrange this equation to get y by itself on the left side, it takes the form y = mx +b. This is called slope intercept form because m is equal to the slope of the line, and b is the value of y when x = 0, which makes it the y-intercept. Converting from slope intercept form to standard form takes little more than basic arithmetic.

TL;DR (Too Long; Didn't Read)

To convert from slope intercept form y = mx + b to standard form Ax + By + C = 0, let m = A/B, collect all terms on the left side of the equation and multiply by the denominator B to get rid of the fraction.

The General Procedure

An equation in slope intercept form has the basic structure y = mx + b.

  1. Subtract mx from both sides

  2. y - mx = (mx - mx ) + b

    y - mx = b

  3. Subtract b from both sides (optional)

  4. y - mx - b = b - b

    y - mx - b = 0

  5. Rearrange to put the x term first

  6. -mx + y - b = 0

  7. Let the fraction A/B represent m

  8. If m is an integer, then B will equal 1.

    -A/Bx + y - b = 0

  9. Multiply both sides of the equation by the denominator B

  10. -Ax + By - Bb = 0

  11. Let Bb = C

  12. -Ax + By - C = 0

Examples:

(1) - The equation of a line in slope intercept form is y = 1/2 x + 5. What is the equation in standard form?

  1. Subtract 1/2 x from both sides of the equation

  2. y - 1/2x = 5

  3. Subtract 5 from both sides

  4. y - 1/2x - 5 = 0

  5. Multiply both sides by the denominator of the fraction

  6. 2y - x - 10 = 0

  7. Rearrange to put x as the first term

  8. -x + 2y - 10 = 0

    You can leave the equation like this, but if you prefer to make x positive, multiply both sides by -1:

    x - 2y + 10 = 0 (or x - 2y = -10)

    (2) - The slope of a line is -3/7 and the y-intercept is 10. What is the equation of the line in standard form?

    The slope intercept form of the line is y = -3/7x + 10. Following the procedure outlined above:

    y + 3/7x - 10 = 0

    7y + 3x - 70 = 0

    3x + 7y -70 = 0 or 3x + 7y = 70

References

About the Author

Chris Deziel holds a Bachelor's degree in physics and a Master's degree in Humanities, He has taught science, math and English at the university level, both in his native Canada and in Japan. He began writing online in 2010, offering information in scientific, cultural and practical topics. His writing covers science, math and home improvement and design, as well as religion and the oriental healing arts.

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