Variable Elimination

${\displaystyle 2x+y=11\,}$

${\displaystyle -4x+3y=13\,}$

Multiply 1st row by 2 aadd to 2nd row

${\displaystyle 4x+2y=22\,}$

${\displaystyle -4x+3y=13\,}$

Yields

${\displaystyle 5y=35\,}$ => ${\displaystyle y=7}$

Multiply 1st row by -3 add to 2nd row

${\displaystyle -6x-3y=-33\,}$

${\displaystyle -4x+3y=13\,}$

Yields

${\displaystyle -10x=-20\,}$ => ${\displaystyle x=2}$

Substitution

If you get a system of equations that looks like this:

${\displaystyle 2x+y=11\,}$

${\displaystyle -4x+3y=13\,}$

You can switch around some terms in the first to get this:

${\displaystyle y=-2x+11\,}$

Then you can substitute that into the bottom one so that it looks like this:

${\displaystyle -4x+3(-2x+11)=13\,}$

${\displaystyle -4x-6x+33=13\,}$

${\displaystyle -10x+33=13\,}$

${\displaystyle -10x=-20\,}$

${\displaystyle x=2\,}$

Then, you can substitute 2 into an x from either equation and solve for y. It's usually easier to substitute it in the one that had the single y. In this case, after substituting 2 for x, you would find that y = 7.

Determinant

If you get a system of equations that looks like this:

${\displaystyle 2x+y=11\,}$

${\displaystyle -4x+3y=13\,}$

Solve for y

${\displaystyle x+{\frac {1}{2}}y={\frac {11}{2}}\,}$

${\displaystyle x+{\frac {3}{-4}}y={\frac {13}{-4}}\,}$

${\displaystyle y({\frac {1}{2}}-{\frac {3}{4}})={\frac {11}{2}}-{\frac {13}{-4}}}$

${\displaystyle y={\frac {{\frac {11}{2}}-{\frac {13}{-4}}}{{\frac {1}{2}}-{\frac {3}{4}}}}}$

Solve for x

${\displaystyle 2x+y=11\,}$

${\displaystyle {\frac {-4}{3}}x+y={\frac {13}{3}}\,}$

${\displaystyle x(2-{\frac {-4}{3}})=11-{\frac {13}{3}}}$

${\displaystyle x={\frac {{\frac {11}{2}}-{\frac {13}{-4}}}{{\frac {1}{2}}-{\frac {3}{4}}}}}$

Find Roots of System of Linear Equations systematically

${\displaystyle A_{1}x+B_{1}y=C_{1}}$

${\displaystyle A_{2}x+B_{2}y=C_{2}}$

Eliminate variable x

${\displaystyle x+{\frac {B_{1}}{A_{1}}}y={\frac {C_{1}}{A_{1}}}}$

${\displaystyle x+{\frac {B_{2}}{A_{2}}}y={\frac {C_{2}}{A_{2}}}}$

Add

${\displaystyle y({\frac {B_{1}}{A_{1}}}-{\frac {B_{2}}{A_{2}}})={\frac {C_{1}}{A_{1}}}-{\frac {C_{2}}{A_{2}}}}$

Solve for y

${\displaystyle y={\frac {{\frac {C_{1}}{A_{1}}}-{\frac {C_{2}}{A_{2}}}}{{\frac {B_{1}}{A_{1}}}-{\frac {B_{2}}{A_{2}}}}}}$

Eliminate variable y

${\displaystyle {\frac {A_{1}}{B_{1}}}x+y={\frac {C_{1}}{B_{1}}}}$

${\displaystyle {\frac {A_{2}}{B_{2}}}x+y={\frac {C_{2}}{B_{2}}}}$

Add

${\displaystyle x({\frac {A_{1}}{B_{1}}}-{\frac {A_{2}}{B_{2}}})={\frac {C_{1}}{B_{1}}}-{\frac {C_{2}}{B_{2}}}}$

Solve for x

${\displaystyle x={\frac {{\frac {C_{1}}{B_{1}}}-{\frac {C_{2}}{B_{2}}}}{{\frac {A_{1}}{B_{1}}}-{\frac {A_{2}}{B_{2}}}}}}$