GENERAL EQUATION OF A LINE WHOSE SLOPE IS EQUAL TO 2/3 - MATHS - 2025

The general equation of a line L is the following: Ax + By + C = 0, where A, B and C are constants, x is the independent variable and y the dependent variable.

The slope of a line, generally denoted by the letter m, that passes through the points P = (x1, y1) and Q = (x0, y0) is the following quotient m: = (y1-y0) / (x1 -x0).

The slope of a line, represents in a way the inclination; More formally, the slope of a line is the tangent of the angle it makes with the X axis.

It should be noted that the order in which the points are named is indifferent, since (y0-y1) / (x0-x1) = - (y1-y0) / (- (x1-x0)) = (y1-y0) / (x1-x0).

Slope of a line

If two points are known through which a line passes, it is easy to calculate its slope. But what if these points are not known?

Given the general equation of a line Ax + By + C = 0, its slope is m = -A / B.

What is the general equation of a line whose slope is 2/3?

As the slope of the line is 2/3 then the equality -A / B = 2/3 is established, with which we can see that A = -2 and B = 3. So the general equation of a line with slope equal to 2/3 is -2x + 3y + C = 0.

It should be clarified that if A = 2 and B = -3 are chosen, the same equation will be obtained. In effect, 2x-3y + C = 0, which is equal to the previous one multiplied by -1. The sign of C does not matter as it is a general constant.

Another observation that can be made is that for A = -4 and B = 6 the same line is obtained, despite the fact that their general equation is different. In this case the general equation is -4x + 6y + C = 0.

Are there other ways to find the general equation of the line?

The answer is yes. If the slope of a line is known, there are two ways, in addition to the previous one, to find the general equation.

For this, the Point-Slope equation and the Shear-Slope equation are used.

-The Point-Slope equation: if m is the slope of a line and P = (x0, y0) a point through which it passes, then the equation y-y0 = m (x-x0) is called the Point-Slope equation.

-The Cut-Slope equation: if m is the slope of a line and (0, b) is the cut of the line with the Y axis, then the equation y = mx + b is called the Cut-Slope equation.

Using the first case, it is obtained that the Point-Slope equation of a line whose slope is 2/3 is given by the expression y-y0 = (2/3) (x-x0).

To arrive at the general equation, multiply by 3 on both sides and group all the terms on one side of the equality, with which we obtain that -2x + 3y + (2 × 0-3y0) = 0 is the general equation of the line, where C = 2 × 0-3y0.

If the second case is used, it is obtained that the equation Cut-Slope of a line whose slope is 2/3 is y = (2/3) x + b.

Again, multiplying by 3 on both sides, and grouping all the variables, we get -2x + 3y-3b = 0. The latter is the general equation of the line where C = -3b.

Actually, looking closely at both cases, it can be seen that the second case is simply a particular case of the first (when x0 = 0).

References

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