# Algebra Challenge: Proving $p,\,q,\,r,\,s,\,t$ Satisfy Equation

• MHB
• anemone
In summary: Now, let's substitute the second equation into the right side of the equation:$(p+q)^2(s^2t^2)+t^2q^2+(s-t)^2(t^2)=2(s^2t^2)(st)+(p+q)^2s^2+(s-t)^2t^2$Using the original equations again, we can simplify this to:$(p+q)^ anemone Gold Member MHB POTW Director Let$p,\,q,\,r,\,s,\,t \in \mathbb {R_+}$satisfying$p^2+pq+q^2=s^2\\ q^2+qr+r^2=t^2\\r^2+rp+p^2=s^2-st+t^2$Prove that$\dfrac{s^2-st+t^2}{s^2t^2}=\dfrac{r^2}{q^2t^2}+\dfrac{p^2}{q^2s^2}-\dfrac{pr}{q^2ts}$Dear forum members, I would like to provide a proof for the given statement. First, we can rewrite the given equations as:$p^2+pq+q^2=s^2\\ q^2+qr+r^2=t^2\\(p^2+q^2+pq)-(st)=t^2-st+s^2$Next, we can rearrange the third equation to get:$(p^2+q^2+pq)-(st)-(t^2-st+s^2)=0\\(p^2+q^2+pq)-(st)-\dfrac{1}{2}((p+q)^2+(s-t)^2)=0$Using this, we can rewrite the original equations as:$p^2+pq+q^2=s^2\\ q^2+qr+r^2=t^2\\(p+q)^2+(s-t)^2=2(st)$Now, let's multiply the first equation by$s^2$, the second equation by$t^2$, and the third equation by$s^2t^2$. We get:$s^2(p^2+pq+q^2)=s^4\\t^2(q^2+qr+r^2)=t^4\\(p+q)^2s^2t^2+(s-t)^2s^2t^2=2(s^2t^2)(st)$Combining these equations, we get:$(p^2+q^2+pq)s^2+(q^2+qr+r^2)t^2+(p+q)^2s^2t^2+(s-t)^2s^2t^2=2(s^2t^2)(st)+s^4+t^4$Using the original equations, we can simplify this to:$(p^2+q^2+pq)s^2+(q^2+qr+r^2)t^2+(p+q)^2s^2t^2+(s-t)^2s^2t^2=2(s^2t^2)(st)+(p+q)^2s^2+(s-t)^2t^2$Next, we can rewrite the left side of this equation as:$(p^2+q^2+pq)s^2+(q^2+qr+r^2)t

## 1. What is the purpose of proving that $p,\,q,\,r,\,s,\,t$ satisfy an equation in algebra?

The purpose of proving that $p,\,q,\,r,\,s,\,t$ satisfy an equation in algebra is to show that the values of these variables make the equation true. This allows us to solve for the unknown variables and find the solution to the equation.

## 2. How do you prove that $p,\,q,\,r,\,s,\,t$ satisfy an equation in algebra?

To prove that $p,\,q,\,r,\,s,\,t$ satisfy an equation in algebra, we substitute the values of these variables into the equation and simplify both sides. If the resulting expressions are equal, then we have proven that the values of $p,\,q,\,r,\,s,\,t$ satisfy the equation.

## 3. Can you give an example of an equation that can be proved using $p,\,q,\,r,\,s,\,t$?

One example of an equation that can be proved using $p,\,q,\,r,\,s,\,t$ is $p^2 + q^2 = r^2 + s^2 + t^2$. We can prove this equation by substituting specific values for $p,\,q,\,r,\,s,\,t$ and showing that the equation holds true.

## 4. What happens if the values of $p,\,q,\,r,\,s,\,t$ do not satisfy the equation?

If the values of $p,\,q,\,r,\,s,\,t$ do not satisfy the equation, then the equation is not true and we cannot solve for the unknown variables. This means that the values of $p,\,q,\,r,\,s,\,t$ do not make the equation true and we need to find different values that will satisfy the equation.

## 5. Why is it important to prove that $p,\,q,\,r,\,s,\,t$ satisfy an equation in algebra?

It is important to prove that $p,\,q,\,r,\,s,\,t$ satisfy an equation in algebra because it allows us to verify the validity of our solution. By proving that the values of $p,\,q,\,r,\,s,\,t$ make the equation true, we can be confident that our solution is correct and that we have solved the equation accurately.

• General Math
Replies
4
Views
2K
• General Math
Replies
2
Views
2K
• General Math
Replies
1
Views
879
• General Math
Replies
2
Views
2K
• General Math
Replies
1
Views
1K
• Precalculus Mathematics Homework Help
Replies
8
Views
750
• General Math
Replies
1
Views
2K
• Math Proof Training and Practice
Replies
61
Views
6K
• General Math
Replies
5
Views
1K
• General Math
Replies
3
Views
2K