Constant Rate of Change

Pairs of Points in a Line Have a Constant Rate of Change

Any linear equation in two variables can written in the following form where y is the dependent variable, x is the independent variable, and m and b can be any real numbers. y=mx+b Now, since this is a linear equation in two variables, all of its solutions lie on a line in the coordinate plane. In what follows, two arbitrary points (x_1,y_1) and (x_2,y_2) will be used, and the rate of change between them will be found by using the rate of change formula. Rate of Change = y_2-y_1/x_2-x_1 Because these arbitrary points are solutions to the linear equation, they satisfy the equation. Therefore, it is possible to find an explicit expression for y_1 and y_2 in terms of x_1 and x_2, respectively. y_1 = mx_1+b y_2 = mx_2+b Now that the explicit form for y_1 and y_2 is known, the rate of change between these points can be calculated.

As stated before, m is a real number and, therefore, a constant. It has been found that, no matter which two points on the line are used, the rate of change between them will always be a constant value. This constant value m is called the slope of the line.

Consecutive Points in a Data Set with Constant Rate of Change Lie on a Line

Consider a set of data points for which the rate of change between every pair of consecutive points is a constant value m. Consider two consecutive points (x_1,y_1) and (x_2,y_2).

Note that since the rate of change between every pair of consecutive points is constant, they all can be rewritten in a similar way.

Next, it will be shown that every point from the data set satisfies the following linear equation in two variables. y-y_1 = m (x-x_1) This will be shown first for the third point from the data set, (x_3,y_3). The expression on the right-hand side of the equation will be rewritten by adding and subtracting y_2 to obtain an identity.

Since (x_3,y_3) and (x_2,y_2), and (x_2,y_2) and (x_1,y_1), are pairs of consecutive points, the differences y_3-y_2 and y_2-y_1 can be rewritten in terms of the constant rate of change m and the corresponding x-values.

Therefore, the third point (x_3,y_3) satisfies the linear equation y-y_1=m(x-x_1). Now it will be shown that the fourth point of the data set, (y_4,x_4), also satisfies the equation.

Recall that it is known that y_4-y_3=m(x_4-x_3) and, from the previous result, it is also known that y_3-y_1 = m (x_3-x_1).

As shown, (x_4,y_4) satisfies the equation as well. This process can be repeated with all the points, and all will satisfy the linear equation y-y_1=m(x-x_1). And since every point satisfies the equation, every point is a solution. Therefore, every point of the data set lies on the line representing this linear equation.