The other day I was messing around on Gbrainy and I was asked a question "which 2 numbers when added together =16, and when multiplied together =55." I know the X and Y are 5 and 11 but I wanted to see if I could algebraically solve it, and after using an entire sheet of paper trying to do so...I couldn't. So my question is for something like this that isn't as easy how could I solve it without knowing that it is 5 and 11? In x+y=16, I know x=16/y but when I plug it back in I get something like 16/y + y = 16, then I multiply the left side by 16 to get 2y=256 and then ultimately y=128. Am I doing something wrong? But still, if given a harder question along these same lines, how could I solve it? I feel like I should be using substitution or a matrix. I even tried doing something like f(x) to replace y with after I solved for it and then just crunching x values but I got pretty lost doing that.
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Our two equations are: $$x + y = 16 \tag{1}$$ $$xy = 55\tag{2}$$ Rewriting equation (1) in terms of just $y =$ something, we get: $$y = 16-x$$ Substituting this into equation (2) leaves us: $$x(16-x) = 55$$ $$16x-x^2=55 \implies x = 5 \ \ \text{or} \ \ 11$$ which can be easily seen by factoring or using the quadratic formula. It follows that $y=11|x=5$ and $y=5|x=11$. Thus your solutions in terms of $(x,y)$ are $(5,11)$ and $(11,5)$. |
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Here is another method: suppose you are told that two numbers, $x$ and $y$, have a certain sum $x+y=S$, and a certain product $xy=P$. How to find $S$ and $P$? We can use the fact that we know how to solve quadratic equations. Notice that $$(t-x)(t-y) = t^2 - (x+y)t + xy = t^2 - St + P.$$ That means that $x$ and $y$ are precisely the solutions to $$t^2 - St + P = 0.$$ In your specific case, $S=16$ and $P=55$. So we want to find the solutions to $$t^2 - 16t + 55 = 0.$$ The quadratic formula gives $$t = \frac{16 \pm\sqrt{256 - 220}}{2} = 8 \pm\frac{1}{2}\sqrt{36} = 8\pm\frac{6}{2} = \left\{\begin{array}{l} 11\\ 5 \end{array}\right.$$ So the two numbers are $5$ and $11$. (Of course, we often solve quadratic equations $t^2 + at + b=0$ by figuring out by eyeballing two numbers whose product is $b$ and whose sum is $-a$, but we can always use the quadratic formula to take the guessing out of it.) |
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We are trying to solve the system of equations $x+y=16$, $xy=55$. Here are a couple of systematic approaches that work in general. Approach $1$: We will use the identity $(x+y)^2-4xy=(x-y)^2$. In our case, we have $(x+y)^2=256$, $4xy=220$, so $(x-y)^2=36$, giving $x-y=\pm 6$. Using $x+y=16$, $x-y=6$, we get by adding that $2x=22$, and therefore $x=11$. It follows that $y=5$. The possibility $x+y=16$, $x-y=-6$ gives nothing new. Adding, we get $2x=10$, so $x=5$, and therefore $y=11$. Approach $2$: From $x+y=16$, we get $y=16-x$. Substitute for $y$ in $xy=55$. We get $x(16-x)=55$. Simplification gives $x^2-16x+55=0$. The quadratic factors as $(x-5)(x-11)$, so our equation becomes $(x-5)(x-11)=0$, which has the solutions $x=5$ and $x=11$. But we cannot necessarily rely on there being such a straightforward factorization. So in general after we get to the stage $x^2-16x+55=0$, we would use the Quadratic Formula. We get
$$x=\frac{16\pm\sqrt{(-16)^2-4(55)}}{2}.$$ Remarks: Recall that the Quadratic Formula says that if $a\ne 0$, then the solutions of the quadratic equation $ax^2+bx+c=0$ are given by $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$$ Your approach was along reasonable lines, but things went wrong in the details. From $xy=55$ we get $x=\frac{55}{y}$. Substituting in the formula $x+y=16$, we get $$\frac{55}{y}+y=16.$$ A reasonable strategy is to multiply through by $y$, getting $55+y^2=16y$, or equivalently $y^2-16y+55=0$. Now we have reached a quadratic equation which is basically the same as the one we reached above. |
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