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anemone
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Assume that $a, b, c, d$ are positive integers and $\dfrac{a}{c}=\dfrac{b}{d}=\dfrac{3}{4}$, and \(\displaystyle \sqrt{a^2+c^2}-\sqrt{b^2+d^2}=15\), find $ac+bd-ad-bc$.
anemone said:Assume that $a, b, c, d$ are positive integers and $\dfrac{a}{c}=\dfrac{b}{d}=\dfrac{3}{4}$, and \(\displaystyle \sqrt{a^2+c^2}-\sqrt{b^2+d^2}=15\), find $ac+bd-ad-bc$.
The value of $b$ cannot be determined from the given information. The equation only provides information about the sum of $ac+bd-ad-bc$, not the individual values of $a, b, c, d$.
To solve for $a$ and $c$, we must use the given information to create a system of equations. From $\dfrac{3}{4}$, we can write the equation $bd-ad=\dfrac{3}{4}$. From $\sqrt{a^2+c^2}=15$, we can square both sides to get $a^2+c^2= 225$. We can then use substitution or elimination methods to solve for $a$ and $c$.
Yes, the given information can result in multiple solutions. This is because the equation $ac+bd-ad-bc$ is a linear equation with two variables, $a$ and $c$. Therefore, there can be infinitely many solutions that satisfy the equation.
The value $\dfrac{3}{4}$ is not significant in terms of solving the equation $ac+bd-ad-bc$. It is simply a given value in the problem. However, it can be used to create an equation and help solve for $a$ and $c$ as shown in the answer to question 2.
No, there is no simpler way to solve this problem. The given information provides two equations with two unknown variables, so a system of equations must be created and solved to find the values of $a$ and $c$.