Standard Form of a Linear Equation

Although slope-intercept form (y = mx + b) is the most common and generally the most useful form of a linear equation, "standard form" refers to a different format. In standard form, the equation is written as ax + by = c, where a, b, and c are constants. For example, 2x + 3y = 6 is in standard form, where a=2, b=3, and c=6.

Since slope-intercept form tells use more about the properites of the line, it's often easier to convert a linear equation from standard form to slope-intercept form. To do this, you need to isolate y on one side of the equation. For example, to convert the equation 2x + 3y = 6 to slope-intercept form, you would first subtract 2x from both sides of the equation to get 3y = -2x + 6. Then, divide every term by 3 to get y = -2/3x + 2. Now that it's in slope-intercept form, we can see that the slope is -2/3 and the y-intercept is 2.

When an equation is in standard form (ax + by = c), the slope can be calculated by the equation -a/b. For example, if the equation is 2x + 3y = 6, the slope is -2/3. You can validate this by converting the equation to slope-intercept form and comparing the slope.