4x^(2)+9y^(2)-16x+18y=11

x+1=2y

asked by guest
on Nov 19, 2024 at 10:32 am



You asked:

Solve the system of equations:\[\begin{aligned}4 \cdot {x}^{2} + 9 \cdot {y}^{2} - 16 x + 18 y &= 11\\x + 1 &= 2 y\end{aligned}\]

MathBot Answer:

The system of equations has one solution.

\[x = \frac{1}{5}, y = \frac{3}{5}\]

Solve \(4 x^{2} - 16 x + 9 y^{2} + 18 y = 11\) for \(x\). \[x = 2 - \frac{3 \sqrt{- \left(y - 1\right) \left(y + 3\right)}}{2}, x = \frac{3 \sqrt{- \left(y - 1\right) \left(y + 3\right)}}{2} + 2\]Substitute \(2 - \frac{3 \sqrt{- \left(y - 1\right) \left(y + 3\right)}}{2}\) for \(x\) in \(x + 1 = 2 y\) and simplify. $$\begin{aligned}x + 1 &amp= 2 y \\ \left(2 - \frac{3 \sqrt{- \left(y - 1\right) \left(y + 3\right)}}{2}\right) + 1 &= 2 y \\ 2 y &= 3 - \frac{3 \sqrt{- \left(y - 1\right) \left(y + 3\right)}}{2} \end{aligned}$$Substitute \(\frac{3}{5}\) into \(4 x^{2} - 16 x + 9 y^{2} + 18 y = 11\) to solve for \(x\). $$\begin{aligned}4 x^{2} - 16 x + 9 \left(\frac{3}{5}\right)^{2} + 18 \cdot \frac{3}{5} &= 11 \\ 4 x^{2} - 16 x + \frac{351}{25} &= 11 \\4 x^{2} - 16 x + \frac{76}{25} &= 0 \\ \frac{4 \cdot \left(5 x - 19\right) \left(5 x - 1\right)}{25} &= 0 \\ x = \frac{1}{5}&, x = \frac{19}{5}\end{aligned}$$This yields the following solution. $$\begin{aligned}x = \frac{1}{5},\,y = \frac{3}{5}\end{aligned}$$

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