- #1
jewel
- 5
- 0
if
x/(b+c-a)=y/(c+a-b)=z/(a+b-c)
prove that..
x(by+cz-ax)=y(cz+ax-by)=z(ax+by-cz)
x/(b+c-a)=y/(c+a-b)=z/(a+b-c)
prove that..
x(by+cz-ax)=y(cz+ax-by)=z(ax+by-cz)
MarkFL said:Can you show us what you have tried?
Showing your work allows our helpers to see where you are stuck and what mistake(s) you may be making, so that they can offer suggestions to get you going again.
Let \(\displaystyle \frac x{b+c-a} = \frac y{c+a-b} = \frac z{a+b-c} = \lambda.\) First show that $x+y+z = \lambda(a+b+c).$ Then see if you can find an expression for $ax$ in terms of $x,y,z$ and $\lambda$. Do the same for $by$ and $cx$. Then you will be able to see if $x(by+cz-ax)$ is the same as $y(cz+ax-by)$ and $z(ax+by-cz)$.jewel said:if
x/(b+c-a)=y/(c+a-b)=z/(a+b-c)
prove that..
x(by+cz-ax)=y(cz+ax-by)=z(ax+by-cz)
thanks for advice... i will try that way and seeOpalg said:Let \(\displaystyle \frac x{b+c-a} = \frac y{c+a-b} = \frac z{a+b-c} = \lambda.\) First show that $x+y+z = \lambda(a+b+c).$ Then see if you can find an expression for $ax$ in terms of $x,y,z$ and $\lambda$. Do the same for $by$ and $cx$. Then you will be able to see if $x(by+cz-ax)$ is the same as $y(cz+ax-by)$ and $z(ax+by-cz)$.
jewel said:thanks for advice... i will try that way and see
There are probably several ways to attack this problem. The method I used was to start with \(\displaystyle \frac x{b+c-a} = \frac y{c+a-b} = \frac z{a+b-c} = \lambda.\) Then $$x = \lambda(b+c-a),$$ $$y = \lambda(c+a-b),$$ $$z = \lambda(a+b-c).$$ Add those equations to get $x+y+z = \lambda(a+b+c)$. Now subtract the first of those three displayed equations, giving $y+z = 2\lambda a$. Next, multiply that by $x$: $$xy+xz = 2\lambda ax.$$ In a similar way, $$yx + yz = 2\lambda by,$$ $$zx+zy = 2\lambda cz.$$ At this stage, take a look at what you are trying to prove, and see if you can get there.Let \(\displaystyle \frac x{b+c-a} = \frac y{c+a-b} = \frac z{a+b-c} = \lambda.\) First show that $x+y+z = \lambda(a+b+c).$ Then see if you can find an expression for $ax$ in terms of $x,y,z$ and $\lambda$. Do the same for $by$ and $cx$. Then you will be able to see if $x(by+cz-ax)$ is the same as $y(cz+ax-by)$ and $z(ax+by-cz)$.
thanks a lot ... i got it ...:)Opalg said:There are probably several ways to attack this problem. The method I used was to start with \(\displaystyle \frac x{b+c-a} = \frac y{c+a-b} = \frac z{a+b-c} = \lambda.\) Then $$x = \lambda(b+c-a),$$ $$y = \lambda(c+a-b),$$ $$z = \lambda(a+b-c).$$ Add those equations to get $x+y+z = \lambda(a+b+c)$. Now subtract the first of those three displayed equations, giving $y+z = 2\lambda a$. Next, multiply that by $x$: $$xy+xz = 2\lambda ax.$$ In a similar way, $$yx + yz = 2\lambda by,$$ $$zx+zy = 2\lambda cz.$$ At this stage, take a look at what you are trying to prove, and see if you can get there.
jewel said:thanks a lot ... i got it ...:)
A ratio is a comparison between two quantities that have the same units. It is expressed in the form of a fraction, where the numerator represents the first quantity and the denominator represents the second quantity.
To prove a ratio given an equation, you need to manipulate the equation algebraically until it is in the form of a ratio. This can be done by dividing both sides of the equation by a common factor or by multiplying both sides by the same number.
There are four types of ratios: part-to-part, part-to-whole, whole-to-part, and whole-to-whole. Part-to-part ratios compare different parts of a whole, while part-to-whole ratios compare a part to the whole. Whole-to-part ratios compare the whole to a part, and whole-to-whole ratios compare two different wholes.
Yes, a ratio can be simplified by dividing both the numerator and denominator by their greatest common factor. This will result in an equivalent ratio that is easier to work with. However, it is important to note that ratios should only be simplified if necessary, as they may be needed in their original form for further calculations.
Ratios are used in many real-life situations, such as in cooking, financial planning, and sports. In cooking, ratios are used to determine the correct proportions of ingredients in a recipe. In financial planning, ratios are used to analyze the financial health of a company. In sports, ratios are used to compare statistics of players or teams.