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Table 15-2 is an S/R derivation that shows how this can be done
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Equations from Graphs
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Let s derive the SI and PS forms of linear equations by looking at how their graphs behave generally Then we ll derive a standard form for a linear equation based on two known points
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Known slope and y -intercept Imagine a line in Cartesian coordinates that has slope m and crosses the y axis at the point (0, b), as shown in Fig 15-6 If we move away from (0, b) on the line, the slope is always equal to the difference in the y value divided by the difference in the x value, or y / x
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Equations from Graphs +y x=0+ x y=b+ y (x,y)
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y x (0,b) m= y x x +x
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Figure 15-6 The SI form of a linear equation can be derived
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from this generic graph
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Suppose we move from (0, b) to some point (x, y) on the line by going x units to the right and y units upward The x coordinate of the point (x, y) will be 0 + x, because we have moved x units horizontally from a point where x = 0 The y coordinate of the point (x, y) will be b + y, because we have moved y units vertically from a point where y = b If we can get an equation that allows us to calculate y in terms of x for the arbitrary point (x, y), then we will have demonstrated how y is a function of x As things turn out, we ll also get the SI form of the equation for the line We can express y in terms of the slope m and the increment x by morphing the formula that defines slope That formula, once again, is m = y / x Multiplying through by x, we get m x = y Now remember that y = b + y We can substitute m x for y in this equation, getting y = b + m x
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246 Graphs of Linear Relations
But in this situation, x is exactly equal to x ! That s because, by traversing the increment x, we have moved from the y axis (where x = 0) horizontally by x units Because of this lucky coincidence, we can substitute x for x in the above equation, getting y = b + mx If we want to be picayune, we can reverse the order of the addends to state it as y = mx + b
Known point and slope Imagine a line in the Cartesian plane that passes through a point whose coordinates are (x0y0), where x0 and y0 are known constants Suppose the line has slope m as shown in Fig 15-7 If we move away from (x0, y0) along the line, the slope is always equal to y / x Let s go from (x0, y0) to some arbitrary point (x, y) on the line, just as we did when we derived the SI equation The x coordinate of (x, y) will be x0 + x, because we have moved x units horizontally from a point where x = x0 The y coordinate of the point (x, y) will be y0 + y, because we have moved y units vertically from a point where y = y0 Now remember, once again, how slope is defined:
m = y / x
+y x = x0 + y = y0 + x y (x,y)
y x (x0 ,y0 ) x +x
Figure 15-7 The PS form of a linear equation can be derived
from this generic graph
Equations from Graphs
As before, we have m x = y Observe that in Fig 15-7, y = y0 + y Let s substitute m x for y here That gives us y = y0 + m x Now we can see from Fig 15-7 that x = x0 + x Subtracting x0 from each side, we obtain x x0 = x Substituting (x x0) for x in the equation for y in terms of y0 and m x, we get y = y0 + m(x x0) Subtracting y0 from each side gets us to the PS form y y0 = m(x x0)