Why can the two equations be added directly to eliminate y in this example?

Conditions

• The system has been reduced to integer coefficients: $3x - 4y = 14$ and $15x + 4y = 22$.
• The equations are aligned term-by-term before addition.

Reasoning, step by step

1. Observe the $y$-coefficients in the reduced system: $-4$ in the first equation and $+4$ in the second.
2. Recognize that $-4$ and $+4$ are additive inverses (opposites).
3. Add the two equations together term by term.
4. The $y$-terms cancel out ($-4y + 4y = 0$), eliminating $y$ from the resulting equation.
5. Solve the remaining equation, $18x = 36$, to find $x = 2$.
6. Note that this direct addition works only because the system was already prepared with opposite coefficients; otherwise, extra manipulation would be needed first.

Example

Adding $3x - 4y = 14$ and $15x + 4y = 22$ gives $(3x+15x) + (-4y+4y) = 14+22$, which simplifies to $18x = 36$, and thus $x = 2$.

Common misconceptions

• Assuming that any linear system can be solved by simply adding the equations as written, regardless of the coefficients.
• Believing that the elimination method always works immediately without needing to check if coefficients are opposites or can be made opposites.

Watch the explanation

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