# Reduce the system of linear equations to upper triangular form and 2x+3y=5 Solve -y=-2+frac{2}{3}x

Question
Forms of linear equations
Reduce the system of linear equations to upper triangular form and
$$2x+3y=5$$
Solve
$$-y=-2+\frac{2}{3}x$$

2020-11-04
We are going to use following elementary row operations:
$$\circ \text{Interchange i-th and j-th row:} R_i \leftrightarrow R_j$$
\circ \text{Multiply i-th row by scalar} \alpha R_i \leftrightarrow \alpha \cdot R_i\)
$$\text{Add} \alpha \text{times i-th row to j-th row: } R_j \rightarrow R_j+\alpha \cdot R_i$$
$$\begin{bmatrix} 2 &3 & 5\\ -\frac{2}{3} & -1 & -2 \end{bmatrix} R_2\rightarrow3 \cdot R_2 \begin{bmatrix} 2 &3 & 5\\ -2 & -3 & -6 \end{bmatrix}$$
$$R_2 \rightarrow R_2+R_1 \begin{bmatrix} 2 &3 & 5\\ 0 & 0 & -1 \end{bmatrix}$$
The second equation is 0=-1 . Contradiction , so the system doesnt have a solution.

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