Math Problem Solver

The system of equations has $2$ solutions.

$$x = \frac{1}{2}, y = 6$$$$x = 2, y = 3$$

Solve $2 x + y = 7$ for $x$. $$x = \frac{7}{2} - \frac{y}{2}$$Substitute $\frac{7}{2} - \frac{y}{2}$ for $x$ in $2 x^{2} + x + 3 y = 19$ and simplify. $$\begin{aligned}2 x^{2} + x + 3 y &= 19 \\ 2 \left(\frac{7}{2} - \frac{y}{2}\right)^{2} + \left(\frac{7}{2} - \frac{y}{2}\right) + 3 y &= 19 \\ y^{2} - 9 y &= -18 \\y^{2} - 9 y + 18 &= 0 \\ \left(y - 6\right) \left(y - 3\right) &= 0 \\ y = 3&, y = 6\end{aligned}$$Substitute $3$ into $2 x + y = 7$ to solve for $x$. $$\begin{aligned}2 x + 3 &= 7\\2 x &= 4\\x &= 2\end{aligned}$$This yields the following solution. $$\begin{aligned}x = 2,\,y = 3\end{aligned}$$Substitute $6$ into $2 x + y = 7$ to solve for $x$. $$\begin{aligned}2 x + 6 &= 7\\2 x &= 1\\x &= \frac{1}{2}\end{aligned}$$This yields the following solution. $$\begin{aligned}x = \frac{1}{2},\,y = 6\end{aligned}$$