Solve the following system of equations simultaneously.

x-y = 1

y^2 −2x=13

asked by guest
on Jan 15, 2025 at 6:37 pm



You asked:

Solve the system of equations:\[\begin{aligned}x - y &= 1\\{y}^{2} - 2 x &= 13\end{aligned}\]

MathBot Answer:

The system of equations has \(2\) solutions.

\[x = -2, y = -3\]\[x = 6, y = 5\]

Solve \(x - y = 1\) for \(x\). \[x = y + 1\]Substitute \(y + 1\) for \(x\) in \(- 2 x + y^{2} = 13\) and simplify. $$\begin{aligned}- 2 x + y^{2} &= 13 \\ - 2 \left(y + 1\right) + y^{2} &= 13 \\ y^{2} - 2 y &= 15 \\y^{2} - 2 y - 15 &= 0 \\ \left(y - 5\right) \left(y + 3\right) &= 0 \\ y = -3&, y = 5\end{aligned}$$Substitute \(-3\) into \(x - y = 1\) to solve for \(x\). \[\begin{aligned}x + 3 &= 1\\x &= -2\end{aligned}\]This yields the following solution. $$\begin{aligned}x = -2,\,y = -3\end{aligned}$$Substitute \(5\) into \(x - y = 1\) to solve for \(x\). \[\begin{aligned}x - 5 &= 1\\x &= 6\end{aligned}\]This yields the following solution. $$\begin{aligned}x = 6,\,y = 5\end{aligned}$$

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