A. Slope-Point Form of a Linear Equation
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A. Slope-Point Form of a Linear Equation
The third form of a linear equation you will look at in detail is the slope-point form. This form comes directly from rearranging the slope formula. Remember that in the slope formula, «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»(«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«mo»,«/mo»«mo»§#160;«/mo»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«mo»)«/mo»«/math» is a point and «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»(«/mo»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«mo»,«/mo»«mo»§#160;«/mo»«msub»«mi»y«/mi»«mn»2«/mn»«/msub»«mo»)«/mo»«/math» is another point.
«math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨16px¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«mi»m«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«msub»«mi»y«/mi»«mn»2«/mn»«/msub»«mo»-«/mo»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«/mrow»«mrow»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«mo»-«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mi»m«/mi»«mo»(«/mo»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«mo»-«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«mo»)«/mo»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«msub»«mi»y«/mi»«mn»2«/mn»«/msub»«mo»-«/mo»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«/mrow»«menclose notation=¨updiagonalstrike¨»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«mo»-«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«/menclose»«/mfrac»«menclose notation=¨updiagonalstrike¨»«mo»(«/mo»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«mo»-«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«mo»)«/mo»«/menclose»«/mtd»«/mtr»«mtr»«mtd»«mi»m«/mi»«mo»(«/mo»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«mo»-«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«mo»)«/mo»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msub»«mi»y«/mi»«mn»2«/mn»«/msub»«mo»-«/mo»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
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Often this form is written using
x instead of
x2 and
y instead of
y2, making the formula «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»m«/mi»«mo»(«/mo»«mi»x«/mi»«mo»-«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«mo»)«/mo»«mo»=«/mo»«mi»y«/mi»«mo»-«/mo»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«/math». The video on the left gives an explanation for the derivation of this form of the equation, as well as how it is used. |
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Slope-Point Form
a linear equation of the form «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»m«/mi»«mo»(«/mo»«mi»x«/mi»«mo»-«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«mo»)«/mo»«mo»=«/mo»«mi»y«/mi»«mo»-«/mo»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«/math», where m represents the slope of the graph of the corresponding linear relation and where «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»(«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«mo»,«/mo»«mo»§#160;«/mo»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«mo»)«/mo»«/math» is a specific point on the graph of the linear relation. |
As the name implies, the slope-point form of a linear equation is most useful when you know the slope, along with a point on the line. Although any point from the graph of the relation can be used with this form, it is often helpful to pick
points that are easy to work with, such as (1,1) or (5,0), if possible.
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Example 1 |
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Determine the slope-point equation of a line with a slope of «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»-«/mo»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/math» that passes through the point (4,−1).
Enter the point and the slope into the slope-point equation. «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨16px¨»«mtable columnspacing=¨0px¨ columnalign=¨right center left¨»«mtr»«mtd»«mi mathcolor=¨#B94A48¨»y«/mi»«mo mathcolor=¨#B94A48¨»-«/mo»«msub mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»y«/mi»«mn»1«/mn»«/msub»«/mtd»«mtd»«mo mathcolor=¨#B94A48¨»=«/mo»«/mtd»«mtd»«mi mathcolor=¨#B94A48¨»m«/mi»«mo mathcolor=¨#B94A48¨»(«/mo»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»-«/mo»«msub mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»x«/mi»«mn»1«/mn»«/msub»«mo mathcolor=¨#B94A48¨»)«/mo»«/mtd»«/mtr»«mtr»«mtd»«mi mathcolor=¨#B94A48¨»y«/mi»«mo mathcolor=¨#B94A48¨»-«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»1«/mn»«mo mathcolor=¨#B94A48¨»)«/mo»«/mtd»«mtd»«mo mathcolor=¨#B94A48¨»=«/mo»«/mtd»«mtd»«mo mathcolor=¨#B94A48¨»-«/mo»«mfrac mathcolor=¨#B94A48¨»«mn»2«/mn»«mn»3«/mn»«/mfrac»«mo mathcolor=¨#B94A48¨»(«/mo»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mo mathcolor=¨#B94A48¨»)«/mo»«/mtd»«/mtr»«mtr»«mtd»«mi mathcolor=¨#B94A48¨»y«/mi»«mo mathcolor=¨#B94A48¨»+«/mo»«mn mathcolor=¨#B94A48¨»1«/mn»«/mtd»«mtd»«mo mathcolor=¨#B94A48¨»=«/mo»«/mtd»«mtd»«mo mathcolor=¨#B94A48¨»-«/mo»«mfrac mathcolor=¨#B94A48¨»«mn»2«/mn»«mn»3«/mn»«/mfrac»«mo mathcolor=¨#B94A48¨»(«/mo»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mo mathcolor=¨#B94A48¨»)«/mo»«/mtd»«/mtr»«/mtable»«/mstyle»«/math» |
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Example 2 |
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The equation «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨16px¨»«mrow»«mi»y«/mi»«mo»-«/mo»«mn»4«/mn»«mo»=«/mo»«mn»2«/mn»«mo».«/mo»«mn»7«/mn»«mo»(«/mo»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«mo»)«/mo»«/mrow»«/mstyle»«/math» represents a linear relation. State the slope of the graph of this relation and a point you know will be on the graph of the relation. The equation is in slope-point form, so the slope can be determined by inspection.
m = 2.7
To interpret this equation correctly, it may help to write the addition as a subtraction of a negative. Doing so will help the equation better resemble the form «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi mathcolor=¨#B94A48¨»y«/mi»«mo mathcolor=¨#B94A48¨»-«/mo»«msub mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»y«/mi»«mn»1«/mn»«/msub»«mo mathcolor=¨#B94A48¨»=«/mo»«mi mathcolor=¨#B94A48¨»m«/mi»«mo mathcolor=¨#B94A48¨»(«/mo»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»-«/mo»«msub mathcolor=¨#B94A48¨»«mi mathcolor=¨#B94A48¨»x«/mi»«mn»1«/mn»«/msub»«mo mathcolor=¨#B94A48¨»)«/mo»«/math». «math style=¨font-family:`Times New Roman`¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨16px¨»«mrow»«mi mathcolor=¨#B94A48¨»y«/mi»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»4«/mn»«mo mathcolor=¨#B94A48¨»=«/mo»«mi mathcolor=¨#B94A48¨»m«/mi»«mo mathcolor=¨#B94A48¨»(«/mo»«mi mathcolor=¨#B94A48¨»x«/mi»«mo mathcolor=¨#B94A48¨»-«/mo»«mo mathcolor=¨#B94A48¨»(«/mo»«mo mathcolor=¨#B94A48¨»-«/mo»«mn mathcolor=¨#B94A48¨»3«/mn»«mo mathcolor=¨#B94A48¨»)«/mo»«mo mathcolor=¨#B94A48¨»)«/mo»«/mrow»«/mstyle»«/math»
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