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# 1.2: 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 $$(5+2)/(3-1)=7/2$$.

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.

1.2: Lines is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by David Guichard.