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Systems of Equations
Lesson 5
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Lesson 5: Solving Systems

Find values satisfying all equations simultaneously.

A system solution is the ordered pair that makes every equation true. Use substitution or elimination to reduce two equations to one variable.

Key formula

{a1x+b1y=c1a2x+b2y=c2\begin{cases}a_1x+b_1y=c_1\\a_2x+b_2y=c_2\end{cases}

Worked example

{−3x+5y=−185x−y=30⇒(x,y)=(6,0)\begin{cases}-3x+5y=-18\\5x-y=30\end{cases}\Rightarrow (x,y)=(6,0)

Checkpoint

Find x in {−x+2y=−21−3x−y=−14\text{Find }x\text{ in }\begin{cases}-x+2y=-21\\-3x-y=-14\end{cases}

Recap

  • - Two equations define one intersection point when independent.
  • - Elimination is efficient with opposite coefficients.
  • - Always verify both values in both equations.
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