Math · 05 · Algebra

Linear Algebra II

Notes

Substitution when a variable is already alone; elimination when coefficients line up.
Elimination: scale one equation so a variable cancels when you add or subtract.
One solution: lines cross. No solution: parallel. Infinitely many: same line.
Slopes of $ax+by=c$ are $-\tfrac{a}{b}$ : compare them to count solutions.
Asked for $x+y$ ? Add or subtract the equations directly.
Desmos: graph both, click the intersection.

One solution

\dfrac{a_1}{a_2}\neq\dfrac{b_1}{b_2}

Different slopes

No solution

\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\neq\dfrac{c_1}{c_2}

Parallel lines

Infinitely many solutions

\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}

Same line

Slope of standard form

ax+by=c\ \Rightarrow\ m=-\dfrac{a}{b}

Subtracting equations but forgetting to subtract every term, including the constant.
Answering $x$ when the question asks for $y$ or $x+y$ .
Calling equal slopes “infinitely many” without checking the intercepts.

“No solution” for constant k: set the slopes equal, solve for k, then confirm the intercepts differ.

10 official SAT questions on Linear Algebra II, easiest first.