Linear equations with integer coefficients

Equality is preserved on both sides

For 4x + 7 = 31, subtract 7 from each side to get 4x = 24. Divide both sides by 4, giving x = 6. Check 4(6) + 7 = 31. The operations undo the addition and multiplication while keeping equal quantities equal.

For 3(x − 2) = 15, divide first to get x − 2 = 5, then add 2: x = 7. Expanding first gives 3x − 6 = 15 and the same solution. A valid method need not use the longest sequence of steps.

The unknown can occur on both sides

For 5x − 4 = 2x + 11, subtract 2x and add 4 to obtain 3x = 15, so x = 5. Substitution gives 21 on both sides. A negative coefficient is also permitted: −2x + 3 = 11 gives −2x = 8 and x = −4.

An equation needs a modelled quantity

A club pays a fixed 6 dollars plus 3 dollars for each badge. A total of 27 dollars gives 6 + 3n = 27, so n = 7 badges. A count must be a non-negative integer and should be checked against the original cost relationship.

Check your understanding 1 Give the operation and the check

Solve -3(x-2) = 12.

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Divide both sides by −3: x − 2 = −4. Add 2: x = −2. Substitution gives −3(−4) = 12. Alternatively expand and apply equal operations on both sides.

No solution is a possible outcome

For 2x + 3 = 2x + 4, subtracting 2x would require 3 = 4, so no value works. For 2x + 3 = 2x + 3, every real x works. An equation is not guaranteed to identify one unique value.