A first-degree equation in one variable reduces to a constant times the unknown plus a constant equal to zero, with a nonzero coefficient on the unknown. The unknown appears to the first power after simplification. A solution is a value that makes both sides equal.
Keep the two sides equal
Adding or subtracting the same expression on both sides preserves the solutions. Multiplying or dividing both sides by the same nonzero constant also preserves them. The shortcut of moving a term and changing its sign is shorthand for subtracting that term on both sides. When variables are in denominators, record the excluded values before transforming the equation.
Solve an equation with parentheses and a variable on each side
In the equation below, distribute the 3 over both terms in parentheses. Subtract the term containing the unknown on the right from both sides. Then add 6 to both sides and divide by the remaining coefficient.
Substitute 5 into the original equation. The left side is 3 times 3, and the right side is 5 plus 4. Both are 9, so 5 is a solution. Checking in the original catches errors in distribution and signs.
When the unknown cancels
If the variable terms cancel and the constants agree, the equation is an identity on its original domain. For example, distributing in twice the sum of x and 1 gives exactly the expression on the other side, twice x plus 2. If the constants disagree, there is no solution. Twice x plus 2 cannot equal twice x plus 3 for any real x.
A coefficient can itself be given by a parameter. In the equation below, a nonzero parameter gives exactly one solution. If the parameter is zero, the equation becomes zero equals six and has no solution. Check such cases before dividing by a parameter.
