Solving Systems by Elimination

System of Linear Equations and Elimination Overview

• Concept 3 focuses on solving systems of linear equations using the elimination method, an algebraic technique used to determine the solution set shared by linear equations.

• A system of linear equations consists of two or more linear equations containing the same set of variables.

• The primary objective of the elimination method is to combine equations algebraically in order to cancel out one variable, producing a single linear equation in terms of the remaining variable.

• The final calculated output is expressed as a coordinate point representing the point of intersection $(x, y)$ on a Cartesian plane.

Step-by-Step Instructions for Solving Systems by Elimination

• Step 1: Write both equations in Standard/Aligned Form

  • Format both equations in standard form:     $Ax + By = C$

  • Ensure that like terms are vertically aligned, placing $x$-terms above $x$-terms, $y$-terms above $y$-terms, equal signs aligned, and constants $C$ aligned on the opposite side.

• Step 2: Multiply one or both equations by numbers to match coefficients

  • Multiply one or both equations by appropriate non-zero numbers so that the coefficients of one variable ($x$ or $y$) are either identical or exact opposites.

  • Identical coefficients share the exact same numeric value and sign.

  • Exact opposite coefficients share the same numeric magnitude but possess opposite algebraic signs.

• Step 3: Add or subtract equations to eliminate that variable

  • Add the two equations together if the matching coefficients are exact opposites.

  • Subtract one equation from the other if the matching coefficients are identical.

  • Carrying out this operation successfully eliminates the targeted variable.

• Step 4: Solve for the remaining variable

  • Isolate and solve for the single remaining variable in the simplified equation obtained from Step 3.

• Step 5: Substitute the found value back into either original equation

  • Take the numerical value calculated in Step 4 and substitute it back into either of the original system equations.

  • Solve the resulting single-variable equation to find the exact value of the second variable.

• Step 6: State your point of intersection

  • Present the final complete solution as an ordered pair coordinate:     $(x, y)$

  • This ordered pair indicates the unique point of intersection where the corresponding linear graphs cross.