Why check the solution in the other original equation rather than the one already used?

Conditions

• A candidate ordered pair has been found.
• At least one original equation remains available for substitution checking.
• The equation used for back-substitution is not reused for the check.

Reasoning, step by step

1. Identify which original equation was used to find the second variable (the first equation, in this case).
2. Select the other original equation (the second equation) for verification.
3. Substitute the found values of $x$ and $y$ into the selected equation.
4. Simplify both sides of the equation using fraction arithmetic.
5. Compare the left-hand side and right-hand side to see if they produce the same numerical value.
6. Conclude that the solution is correct if the resulting statement is a true equality.

Example

Since $y=-2$ was found by substituting $x=2$ into the first equation $\frac{1}{2}x - \frac{2}{3}y = \frac{7}{3}$, the instructor checks the pair $(2, -2)$ in the second equation $\frac{3}{4}x + \frac{1}{5}y = \frac{11}{10}$. Substituting gives $\frac{3}{4}(2) + \frac{1}{5}(-2) = \frac{6}{4} - \frac{2}{5} = \frac{30}{20} - \frac{8}{20} = \frac{22}{20} = \frac{11}{10}$, which matches the right-hand side.

Common misconceptions

• Assuming that finding a solution by substitution automatically guarantees it satisfies the other equation without verification.
• Checking the solution in the same equation used to derive it, which would always produce a tautology and not validate the system.

Watch the explanation

Explore next

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• What is the final solution to the system?