Linear equations: preserve the balance
Turn a word problem into an equation, solve it with reversible steps and check the answer in context.
In this guide
An equation is a condition to satisfy
An expression such as 4x + 7 names a quantity. An equation such as 4x + 7 = 31 states that two quantities are equal. Solving means finding values that make that statement true. Adding or subtracting the same amount on both sides preserves equality. Multiplying or dividing both sides by the same nonzero constant also preserves equality. These are operations on both sides of the equation, not a rule that terms mysteriously change sign when moved across a line.
Worked example: variables on both sides
Solve 5(x - 2) + 4 = 2x + 15. Distribute the five to both terms inside the parentheses: 5x - 10 + 4 = 2x + 15. Combine constants to get 5x - 6 = 2x + 15. Subtract 2x from both sides: 3x - 6 = 15. Add six to both sides: 3x = 21. Divide by three: x = 7. Check in the original equation: the left side is 5(7 - 2) + 4 = 29; the right side is 2(7) + 15 = 29.
Translate the relationship before manipulating symbols
A fictional event charges a PHP 120 group booking fee plus PHP 35 for each participant. The total is PHP 540. Let n be the number of participants. The relationship is 120 + 35n = 540, because the fixed fee occurs once and the per-person charge occurs n times. Subtracting 120 gives 35n = 420; dividing by 35 gives n = 12. Substitute back: 120 + 35(12) = 540. A negative or fractional participant count would require checking the model or interpreting a constraint.
Fractions and unusual outcomes
For x/4 + 3 = 8, subtract three to get x/4 = 5, then multiply by four to get x = 20. If an equation simplifies to 0 = 0, every allowed value satisfies it: for example, 2(x + 1) = 2x + 2 is an identity. If it simplifies to 0 = 5, no value satisfies it. An equation is not guaranteed to have one numerical answer just because it contains a variable. Domain restrictions, such as excluding a zero denominator, still apply.
Avoid these shortcuts
Apply a negative sign to every term inside parentheses: -(x - 4) equals -x + 4. Keep an equals sign between expressions that are actually equal. Writing a long chain of unrelated intermediate numbers can conceal an invalid step. When checking, use the original equation rather than a later line that might already contain your mistake. For word problems, state what the variable represents and include the appropriate unit in the conclusion.
- Distribute multiplication across every term in a sum or difference.
- Combine only like terms; 3x + 2 is not 5x.
- Do not divide by an expression that might be zero without considering that case.
Try it yourself
Solve 3(2x + 1) - 5 = 4x + 10 and verify your result by substitution. A separate service has a PHP 75 setup fee and costs PHP 18 per item. How many items are on a PHP 291 bill? Explain why dividing the whole bill by 18 would be incorrect.
Reveal the worked answer
Solve 3(2x + 1) - 5 = 4x + 10. Expanding gives 6x - 2 = 4x + 10, then 2x = 12 and x = 6. Substitution gives 34 on both sides. For a second check, a service has a PHP 75 setup fee and costs PHP 18 per item. A PHP 291 bill represents (291 - 75)/18 = 12 items. Explain why dividing 291 by 18 immediately would be incorrect: that treats the fixed setup fee as though it were part of the item count.
Solve 3(2x + 1) - 5 = 4x + 10. Expanding gives 6x - 2 = 4x + 10, then 2x = 12 and x = 6. Substitution gives 34 on both sides. For a second check, a service has a PHP 75 setup fee and costs PHP 18 per item. A PHP 291 bill represents (291 - 75)/18 = 12 items. Explain why dividing 291 by 18 immediately would be incorrect: that treats the fixed setup fee as though it were part of the item count.
Put one idea into practice.
Try a short mathematics session, then use the explanations to choose what to review next.
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