Keep each operation balanced across the equation, then check the result in the original statement and the context.
A linear equation in one variable can usually be solved by distributing, collecting like terms, and isolating the variable. The main risks are dropping a sign, applying an operation to only one term, or solving for a quantity different from the one requested.
Worked example: parentheses and negative signs
All examples are original teaching exercises, not released SAT questions. Solve 3(2x − 1) − 4 = 2x + 9. Distribute to obtain 6x − 3 − 4 = 2x + 9, so 4x = 16 and x = 4.
The original left side is 3(8 − 1) − 4 = 17; the right side is 8 + 9 = 17. Checking the original catches a distribution error that checking only the simplified line would miss.
Worked example: clear fractions across every term
For (x − 2)/3 + x/2 = 6, multiply the entire equation by 6: 2(x − 2) + 3x = 36. Then 5x − 4 = 36, so x = 8. In the original equation, 6/3 + 8/2 = 2 + 4 = 6.
The multiplier applies to the right side too. Do not cancel a denominator across an addition without multiplying all the terms.
Worked example: fixed and variable pay
A worker earns $18 per hour plus a one-time $24 bonus. For total pay of $150, the equation is 18h + 24 = 150. Subtract the bonus before dividing: h = 126/18 = 7 hours. Dividing the entire pay by 18 would wrongly treat the bonus as hourly pay.
An equation may have no solution or infinitely many
The equation 2(x + 3) = 2x + 6 simplifies to 6 = 6. Every real x satisfies it. The equation 2(x + 3) = 2x + 7 simplifies to 6 = 7, which is impossible, so there is no solution.
If a coefficient contains a parameter, inspect zero before dividing. For (k − 2)x = 6, k ≠ 2 gives x = 6/(k − 2), but k = 2 gives 0 = 6 and no solution. Division by k − 2 would hide that case.
Answer the expression the question asks for
If solving gives x = 4 but the question asks for 3x − 2, the answer is 10. For a length, time, or count, check the stated restrictions and include the appropriate unit in your reasoning.
In practice, log distribution, fraction, context, solution-count, and final-expression errors. For equations involving two changing quantities, use linear equations in two variables.
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