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# 1.1: Lines

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If we have two points $$A(x_1,y_1)$$ and $$B(x_2,y_2)$$, then we can draw one and only one line through both points. By the slope of this line we mean the ratio of $$\Delta y$$ to $$\Delta x$$. The slope is often denoted $$m$$: $$m=\Delta y/\Delta x=(y_2-y_1)/(x_2-x_1)$$. For example, the line joining the points $$(1,-2)$$ and $$(3,5)$$ has slope $$(5+2)/(3-1)=7/2$$.

Example 1.1.1

According to the 1990 U.S. federal income tax schedules, a head of household paid 15% on taxable income up to $26050. If taxable income was between$26050 and $134930, then, in addition, 28% was to be paid on the amount between$26050 and $67200, and 33% paid on the amount over$67200 (if any). Interpret the tax bracket information (15%, 28%, or 33%) using mathematical terminology, and graph the tax on the $$y$$-axis against the taxable income on the $$x$$-axis.

Solution

The percentages, when converted to decimal values 0.15, 0.28, and 0.33, are the slopes of the straight lines which form the graph of the tax for the corresponding tax brackets. The tax graph is what's called apolygonal line, i.e., it's made up of several straight line segments of different slopes. The first line starts at the point (0,0) and heads upward with slope 0.15 (i.e., it goes upward 15 for every increase of 100 in the $$x$$-direction), until it reaches the point above $$x=26050$$. Then the graph "bends upward,'' i.e., the slope changes to 0.28. As the horizontal coordinate goes from $$x=26050$$ to $$x=67200$$, the line goes upward 28 for each 100 in the $$x$$-direction. At $$x=67200$$ the line turns upward again and continues with slope 0.33. See figure 1.1.1.

Figure 1.1.1. Tax vs. income.

The most familiar form of the equation of a straight line is: $$y=mx+b$$. Here $$m$$ is the slope of the line: if you increase $$x$$ by 1, the equation tells you that you have to increase $$y$$ by $$m$$. If you increase $$x$$ by $$\Delta x$$, then $$y$$ increases by $$\Delta y=m\Delta x$$. The number $$b$$ is called the y-intercept, because it is where the line crosses the $$y$$-axis. If you know two points on a line, the formula $$m=(y_2-y_1)/ (x_2-x_1)$$ gives you the slope. Once you know a point and the slope, then the $$y$$-intercept can be found by substituting the coordinates of either point in the equation: $$y_1=mx_1+b$$, i.e., $$b=y_1-mx_1$$. Alternatively, one can use the "point-slope'' form of the equation of a straight line: start with $$(y-y_1)/(x-x_1)=m$$ and then multiply to get $$(y-y_1)=m(x-x_1)$$, the point-slope form. Of course, this may be further manipulated to get $$y=mx-mx_1+y_1$$, which is essentially the "$$mx+b$$'' form.