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# algebra

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Find all pairs (x,y) of real numbers such that x + y = 10 and x^2 + y^2 = 56 + xy.

Feb 4, 2021

#1
+1

This is a system of equations. Let's try and solve it.

While it is possible to solve for x or y and then do direct substitution into the remaining equation, I noticed a possible shortcut that is worth knowing for future algebraic problems like this one. This may not save too much time for this particular problem, but I figured it would be to your benefit if I showed this strange substitution to you.

$$\fbox{1}\; x + y = 10\\ (x+y)^2 = 10^2\\ x^2 + 2xy + y^2 = 100\\ {\color{red}x^2 + y^2} = 100 - 2xy$$

I squared equation 1 and rearranged a few terms. To the untrained eye, this may seem to complicate matters unnecessarily, but I am setting up a creative substitution.

$$\fbox{2}\; {\color{red} x^2 + y^2} = 56 + xy\\ 100 - 2xy = 56 + xy\\ 3xy = 44\\ y = \frac{44}{3x}$$

Let's use this substitution in equation 1 and solve it.

$$\fbox{1}\; x+y = 10\\ x+\frac{44}{3x}=10\\ 3x^2+44=30x\\ 3x^2-30x+44=0$$

This quadratic is neither easy nor factorable. I will result to the quadratic formula for this particular question.

$$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a};a=3, b=-30, c=44\\ x_{1,2}=\frac{30\pm\sqrt{900-528}}{6}\\ x_{1,2}=\frac{30\pm\sqrt{372}}{6}\\ x_{1,2}=\frac{30\pm2\sqrt{93}}{6}\\ x_1=\frac{15-\sqrt{93}}{3}\text{ and }x_2=\frac{15+\sqrt{93}}{3}$$

Let's find the corresponding y-values by substituting these values for x into $$y=\frac{44}{3x}$$.

 $$y_1=\frac{44}{3x_1}\\ y_1=\frac{44}{3*\frac{15-\sqrt{93}}{3}}\\ y_1=\frac{44}{15-\sqrt{93}}$$ $$y_2=\frac{44}{3*\frac{15+\sqrt{93}}{3}}\\ y_2=\frac{44}{15+\sqrt{93}}$$

There are two coordinates of intersections. One is at $$\left(\frac{15-\sqrt{93}}{3},\frac{44}{15-\sqrt{93}} \right)$$, and the other is at $$\left(\frac{15+\sqrt{93}}{3}, \frac{44}{15+\sqrt{93}} \right)$$

Feb 4, 2021
#2
+118470
+1

x +  y  =10    ⇒   y  =10 - x

So

x^2 + y^2  =   56 +  xy

x^2  +  (10 - x)^2  =  56  + x ( 10- x)

x^2  + x^2    -20x  + 100  =  56  + 10x  - x^2

3x^2  - 30x  + 44  =  0

Q Formula

x  =   30  ±  sqrt  [ 30^2  -  4 * 3 * 44 ]                 30 ± sqrt (372)

___________________________  =          ___________     =

2 *  3                                                      6

30 ±  2sqrt (93)

____________

6

So       x  =   one  of these  solutions  and  y = the other

Feb 4, 2021