Elimination and Equivalent Systems
Aligned to A-REI.C.5 — Common Core State Standards for Mathematics.
What this lesson teaches
In a system, replacing one equation with the sum of it and a multiple of the other does not change the solution set — this is why the elimination method works.
Worked examples
Example 1
Solve x + y = 6 and x − y = 2. Add the equations: 2x = 8, so x = 4. Substitute into x + y = 6: 4 + y = 6, so y = 2. The solution is (4, 2).
Example 2
Solve 3x + 2y = 16 and x − y = 2. Multiply the second by 2: 2x − 2y = 4. Add: 5x = 20, x = 4, y = 2.
Example 3
Adding a multiple of one equation to another does not change the solution set. That is the whole justification for elimination.
Practice questions
- Solve x + y = 10 and x − y = 4 by adding the equations.
- Solve 2x + y = 7 and x − y = 2 by adding to eliminate y.
- Explain why adding a multiple of one equation to another keeps the same solution.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Eliminación y Sistemas Equivalentes
En un sistema, reemplazar una ecuación por la suma de ella y un múltiplo de la otra no cambia el conjunto solución: por eso funciona el método de eliminación.
Ejemplo 1: Resuelve x + y = 6 y x − y = 2. Suma las ecuaciones: 2x = 8, así que x = 4. Sustituye en x + y = 6: 4 + y = 6, así que y = 2. La solución es (4, 2).
Ejemplo 2: Resuelve 3x + 2y = 16 y x − y = 2. Multiplica la segunda por 2: 2x − 2y = 4. Suma: 5x = 20, x = 4, y = 2.
Ejemplo 3: Sumar un múltiplo de una ecuación a otra no cambia el conjunto solución. Esa es toda la justificación de la eliminación.
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