solve the simultaneous equations x^2-xy+y^2=7, 2x-y = 5

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on Jan 30, 2025 at 12:50 am



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Solve the system of equations:x2xy+y2=72xy=5\begin{aligned}{x}^{2} - x y + {y}^{2} &= 7\\2 x - y &= 5\end{aligned}

MathBot Answer:

The system of equations has 22 solutions.

x=2,y=1x = 2, y = -1x=3,y=1x = 3, y = 1


Solve x2xy+y2=7x^{2} - x y + y^{2} = 7 for xx. x=y2283y22,x=y2+283y22x = \frac{y}{2} - \frac{\sqrt{28 - 3 y^{2}}}{2}, x = \frac{y}{2} + \frac{\sqrt{28 - 3 y^{2}}}{2}Substitute y2283y22\frac{y}{2} - \frac{\sqrt{28 - 3 y^{2}}}{2} for xx in 2xy=52 x - y = 5 and simplify. 2xy=52(y2283y22)y=5283y2=5\begin{aligned}2 x - y &= 5 \\ 2 \left(\frac{y}{2} - \frac{\sqrt{28 - 3 y^{2}}}{2}\right) - y &= 5 \\ \sqrt{28 - 3 y^{2}} &= -5 \end{aligned}Substitute y2283y22\frac{y}{2} - \frac{\sqrt{28 - 3 y^{2}}}{2} for xx in 2xy=52 x - y = 5 and simplify. 2xy=52(y2283y22)y=5283y2=5\begin{aligned}2 x - y &= 5 \\ 2 \left(\frac{y}{2} - \frac{\sqrt{28 - 3 y^{2}}}{2}\right) - y &= 5 \\ \sqrt{28 - 3 y^{2}} &= -5 \end{aligned}Substitute y2+283y22\frac{y}{2} + \frac{\sqrt{28 - 3 y^{2}}}{2} for xx in 2xy=52 x - y = 5 and simplify. 2xy=52(y2+283y22)y=5283y2=5\begin{aligned}2 x - y &= 5 \\ 2 \left(\frac{y}{2} + \frac{\sqrt{28 - 3 y^{2}}}{2}\right) - y &= 5 \\ \sqrt{28 - 3 y^{2}} &= 5 \end{aligned}Substitute 1-1 into x2xy+y2=7x^{2} - x y + y^{2} = 7 to solve for xx. x2x(1)+(1)2=7x2+x+1=7x2+x6=0(x2)(x+3)=0x=3,x=2\begin{aligned}x^{2} - x \left(-1\right) + \left(-1\right)^{2} &= 7 \\ x^{2} + x + 1 &= 7 \\x^{2} + x - 6 &= 0 \\ \left(x - 2\right) \left(x + 3\right) &= 0 \\ x = -3&, x = 2\end{aligned}This yields the following solution. x=2,y=1\begin{aligned}x = 2,\,y = -1\end{aligned}Substitute 11 into x2xy+y2=7x^{2} - x y + y^{2} = 7 to solve for xx. x2x1+12=7x2x+1=7x2x6=0(x3)(x+2)=0x=2,x=3\begin{aligned}x^{2} - x 1 + 1^{2} &= 7 \\ x^{2} - x + 1 &= 7 \\x^{2} - x - 6 &= 0 \\ \left(x - 3\right) \left(x + 2\right) &= 0 \\ x = -2&, x = 3\end{aligned}This yields the following solution. x=3,y=1\begin{aligned}x = 3,\,y = 1\end{aligned}

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