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What is the coefficient of the X-term of the equation below, when the equation is written in standard form?

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3 Answers

Solution: A = 2   Please read the following step by step solution.

STEP 1:  Read, understand the situation within, identify and pull out important information.

  • y = (2/3)x - 6  it's in a slope form equation y=mx+b,  where m(slope)=2/3;  b(y-intercept)=-6
  • The standard form equation leads to the slope form equation just by isolating "y" and by doing opposite operations.
  • To transform a "slope form" into a "standard form", we just do the inverse process: the variables x, y, must be in the same side of the equation.

STEP 2: Translate keywords to their mathematical symbols:

  • Standard form equation:  Ax + By = C,  where A is a positive integer, B and C integer numbers
  • A = ? ;  B = ? ; C = ?

STEP 3: Set up and solve the equation or problem:

               y = (2/3)x - 6    Get rid of the variable x (with its coefficient) on the right side

       -(2/3)x   -(2/3)x

-(2/3)x + y = -6            As "A" must be positive integer, get first rid of the denominator of the fraction coefficient of "x", by multiplying the whole equation by "3"

              3 * ((-2/3) (x) + y = -6)        Apply the distributive property of multiplication

         (3/1)*(-2/3) (x) + 3y = 3*(-6)     

      (3*-2)/(1*3) (x) + 3y   = -18        Simplify

                        -2x + 3y    = -18     Now get rid of the negative sign of "x" by multiplying the whole equation by -1

                   (-1) (-2x + 3y = -18)   Apply the distributive property of multiplication

                            2x  - 3y = 18     There you have the equation in standard form!  A = 2

STEP 4: Check the solution:

Transform the standard form equation solution into a slope intercept form. We must get the original equation given in slope intercept form!

       2x - 3y = 18     Isolate "y" by first getting rid of 2x in the left side of the equation

      -2x          -2x

            -3y = -2x + 18      Get rid of the coefficient -3, by dividing the whole equation by -3

 (-1/3) (-3y = -2x + 18)    Applying the distributive property of multiplication

       (3/3)y = (2/3)x + (-1/3) (18)    Simplifying

y = (2/3)x -6   We got the same original slope intercept form equation!

STEP 5: Curiosities

  • Having the equation in "standard form" it's really easy to get the y-intercept point (0,y) and the x-intercept point (x,0).

          y-intercept:  2(0)-3y=18; -3y=18; (-3/-3)y=(18)/(-3); y=-6;

          y-intercept point (0, -6)

          x-intercept:  2x-3(0)=18; 2x=18; (2/2)x=(18)/(2); x=9;

          x-intercept point (9, 0)

  • Graphing the equation in standard form: find the y-intercept point and x-intercept point, as we  already did it. Plot them in a Cartesian plane and draw a line through both intercept points. And voila' you got it!

 

 

Standard form of a linear equations is Ax + By = C.

y = 2/3x - 6  

Multiply entire equation by 3 to remove the fraction.

3y = 2x - 18

Next, subtract 2x from both sides of the equation.

-2x + 3y = -18

Now we have the equation in standard form therefore the coefficient of the x-term is -2.

Note: There are infinite standard forms.

Standard form of a linear equation is Ax+By=C, where A, B, and C are all integers.  Your equation is given in slope-intercept form (usually stated as y=mx+b), so we have to isolate the constant term and then multiply through by the least common denominator of any fractions in the coefficients.

y = (2/3)x + (-6)

(-2/3)x + y = (-6)

(-2)x + 3y = (-18)

And the coefficient of x is -2.