线性方程组与消元

Introduction: From One Unknown to Two

Linear Systems and the Elimination Lemma

After replacing the equation $F$ by $E+cF$ ($c\neq 0$), what happens to the solution set?

The Determinant D and Three Cases

For $\begin{cases}ax+by=e\\cx+dy=f\end{cases}$, when $D=ad-bc\neq 0$, which case does the system fall into?

Further Check When D=0

When $D=0$ and elimination yields the identity $0=0$, what is the relationship between the two lines?

Cramer's Rule

What is the expression for $x$ in Cramer's rule?

Worked Example: Solving a Two-Variable System by Elimination

Solve $\begin{cases}2x+y=7\\x-y=2\end{cases}$. Write the answer as $x;y$ (no spaces).

For $\begin{cases}2x+3y=8\\4x+6y=15\end{cases}$, compute $D=ad-bc$ and classify which case the system falls into. Write the answer as $D$;determined/contradictory/indeterminate.

Use Cramer's rule to solve $\begin{cases}3x+2y=11\\x-y=2\end{cases}$. Write the answer as $x;y$ (no spaces).

How does $D=ad-bc$ distinguish the three cases?

Explain in your own words: why does replacing the equation $F$ by $E+cF$ leave the solution set of the system unchanged? What is the key to the proof?

Comprehensive Check

Correcting Common Errors and Lesson Summary