## Precalculus (10th Edition)

Consistent Solution set: $\left\{\left(x,y\right)|x=4-2y,y\text{ is any real number}\right\}$
We are given the system of equations: $\begin{cases} x+2y=4\\ 2x+4y=8 \end{cases}$ Write the augmented matrix: $\begin{bmatrix}1&2&|&4\\2&4&|&8\end{bmatrix}$ Perform row operations to bring the matrix to the reduced row echelon form: $R_1=\dfrac{1}{2}r_1$ $\begin{bmatrix}1&2&|&4\\1&2&|&4\end{bmatrix}$ $R_1=-r_1+r_2$ $\begin{bmatrix}1&2&|&4\\0&0&|&0\end{bmatrix}$ The last row only contains zeroes, so the system is consistent, having infinitely many solutions. Write the corresponding system of equations: $\begin{cases} x+2y=4\\ 0=0 \end{cases}$ Express $x$ in terms of $y$: $x+2y=4\Rightarrow x=4-2y$ The solution set is: $\left\{\left(x,y\right)|x=4-2y,y\text{ is any real number}\right\}$