Forms of equation

Forms of equation

General (or standard) form

In the general (or standard) form the linear equation is written as:

where A and B are not both equal to zero. The equation is usually written so that A ≥ 0, by convention. The graph of the equation is a straight line, and every straight line can be represented by an equation in the above form. If A is nonzero, then the x-intercept, that is, the x-coordinate of the point where the graph crosses the x-axis (where, y is zero), is C/A. If B is nonzero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis (where x is zero), is C/B, and the slope of the line is −A/B. The general form is sometimes written as:

where a and b are not both equal to zero. The two versions can be converted from one to the other by moving the constant term to the other side of the equal sign.

Video: Linear Equations in Standard Form

Slope–intercept form

where m is the slope of the line and b is the y intercept, which is the y coordinate of the location where the line crosses the y axis. This can be seen by letting x = 0, which immediately gives y = b. It may be helpful to think about this in terms of y = b + mx; where the line passes through the point (0, b) and extends to the left and right at a slope of m. Vertical lines, having undefined slope, cannot be represented by this form.

A corresponding form exists for the x intercept, though it is less-used, since y is conventionally a function of x:

Analogously, horizontal lines cannot be represented in this form. If a line is neither horizontal nor vertical, it can be expressed in both these forms, with , so . Expressing y as a function of xgives the form:

which is equivalent to the polynomial factorization of the y intercept form. This is useful when the x intercept is of more interest than the y intercept. Expanding both forms shows that , so , expressing the x intercept in terms of the y intercept and slope, or conversely.

Video: Slope Intercept Form  

Point–slope form

where m is the slope of the line and (x₁,y₁) is any point on the line.

The point-slope form expresses the fact that the difference in the y coordinate between two points on a line (that is, y − y₁) is proportional to the difference in the x coordinate (that is, x − x₁). The proportionality constant is m (the slope of the line).

Video: Point Slope Form

Two-point form

where (x₁, y₁) and (x₂, y₂) are two points on the line with x₂ ≠ x₁. This is equivalent to the point-slope form above, where the slope is explicitly given as (y₂ − y₁)/(x₂ − x₁).

Multiplying both sides of this equation by (x₂ − x₁) yields a form of the line generally referred to as the symmetric form:

Expanding the products and regrouping the terms leads to the general form:

Using a determinant, one gets a determinant form, easy to remember:

Video: Introduction To Two Point Form 

Intercept form

where a and b must be nonzero. The graph of the .....................

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