Lösung 2.2:5d
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If two non-vertical lines are perpendicular to each other, their slopes <math>k_{1}</math> and <math>k_{2}</math> satisfy the relation <math>k_{1}k_{2}=-1</math>, and from this we have that the line we are looking for must have a slope that is given by | If two non-vertical lines are perpendicular to each other, their slopes <math>k_{1}</math> and <math>k_{2}</math> satisfy the relation <math>k_{1}k_{2}=-1</math>, and from this we have that the line we are looking for must have a slope that is given by | ||
| - | {{ | + | {{Abgesetzte Formel||<math>k_{2} = -\frac{1}{k_{1}} = -\frac{1}{2}</math>}} |
since the line <math>y=2x+5</math> has a slope <math>k_{1}=2</math> | since the line <math>y=2x+5</math> has a slope <math>k_{1}=2</math> | ||
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The line we are looking for can thus be written in the form | The line we are looking for can thus be written in the form | ||
| - | {{ | + | {{Abgesetzte Formel||<math>y=-\frac{1}{2}x+m</math>}} |
with ''m'' as an unknown constant. | with ''m'' as an unknown constant. | ||
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Because the point (2,4) should lie on the line, (2,4) must satisfy the equation of the line, | Because the point (2,4) should lie on the line, (2,4) must satisfy the equation of the line, | ||
| - | {{ | + | {{Abgesetzte Formel||<math>4=-\frac{1}{2}\cdot 2+m\,,</math>}} |
i.e. <math>m=5</math>. The equation of the line is <math>y=-\frac{1}{2}x+5</math>. | i.e. <math>m=5</math>. The equation of the line is <math>y=-\frac{1}{2}x+5</math>. | ||
Version vom 08:28, 22. Okt. 2008
If two non-vertical lines are perpendicular to each other, their slopes \displaystyle k_{1} and \displaystyle k_{2} satisfy the relation \displaystyle k_{1}k_{2}=-1, and from this we have that the line we are looking for must have a slope that is given by
| \displaystyle k_{2} = -\frac{1}{k_{1}} = -\frac{1}{2} |
since the line \displaystyle y=2x+5 has a slope \displaystyle k_{1}=2 (the coefficient in front of x).
The line we are looking for can thus be written in the form
| \displaystyle y=-\frac{1}{2}x+m |
with m as an unknown constant.
Because the point (2,4) should lie on the line, (2,4) must satisfy the equation of the line,
| \displaystyle 4=-\frac{1}{2}\cdot 2+m\,, |
i.e. \displaystyle m=5. The equation of the line is \displaystyle y=-\frac{1}{2}x+5.
