1. to be able to write the equation of a line in standard form.
Standard Form: the standard form of a line is in the form Ax + By = C where A is a positive integer, and B, and C are integers.
The standard form of a line is just another way of writing the equation of a line. It gives all of the same information as the slope-intercept form that we learned about on Day 5 just written differently.
Recall that the slope-intercept form of a line is: y = mx + b. To change this into standard form, we start by moving the x-term to the left side of the equation. This is done by subtracting mx from both sides. We now have the equation, -mx + y = b. The coefficient of the x-term should be a positive integer value, so we multiply the entire equation by an integer value that will make the coefficient positive, as well as, all of the coefficeints integers. This gives us the standard form: Ax + By = C.
Example1: Write the equation of the line: y = -3x + 6 in standard form.
First, we need to move the x-term to the left side of the equation so we add 3x to both sides. Doing this gives us: 3x + y = 6. Here, the coefficient of the x-term is a positive integers and all other values are integers, so we are done.
Again, start by moving the x-term to the left. Subtract 2x from both sides to get: -2x + y = 7. We need the x-term to be positive, so multiply the equation by -1 to get our answer: 2x - y = 7.
Example3: Write the equation of the line: y = (1/2)x + 8 in standard form.
First, subtract (1/2)x form both sides to get: (-1/2)x + y = 8. Here, we need to get rid of the (-1/2) by multiply by its reciprocal, -2. Doing this gives us: x - 2y = -16 and we are done.
Example4: Write the equation of the line with a slope of (-3/4 ) that passes through the point (0,6) in standard form.
First, we have to write the equation of a line using the given information. We know m = (-3/4) and b = 6, so we use slope-intercept form, y = mx + b to start. Substitution gives us the equation of the line as: y = (-3/4)x + 6. Now, we must convert to standard form. First add (3/4)x to both sides to get: (3/4)x + y = 6. Finally, we must get rid of the fraction so, we clear the fraction by multiplying by the common denominator of all of the terms which is 4. This multiplication yields the answer which is: 3x + 4y = 24.
Note: I have seen it where fractions have been allowed to stay in standard form. In particular, our book would not have cleared the fraction in example 4. The authors would have left the answer as: (3/4)x + y = 6. However, for our class, we will clear the fractions. It is a very useful skill that will come in handy later in the year.
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