Solving Equations

Solving an equation means finding the value of the unknown that makes it true. In x + 3 = 8, the unknown is x; when x = 5 the equation reads 5 + 3 = 8, which is correct, so x = 5 is the solution. A linear equation like this has exactly one solution, and you can always test it by putting the value back in.

Three steps. Simplify both sides: expand brackets and collect like terms. Rearrange: get the x terms on one side and the numbers on the other by doing the same thing to both sides. Divide by the number in front of x. For 3x + 2 = 11: subtract 2 from both sides to get 3x = 9, then divide by 3 to get x = 3.

Because an equation is a balance: the two sides are equal, and they only stay equal if you change both in the same way. In 2x − 4 = 6, adding 4 to both sides gives 2x = 10, and dividing both sides by 2 gives x = 5. Change one side only and the sides no longer match, so the answer would be wrong.

Expand the brackets first, then carry on as normal. For 2(3x + 5) − 4 = 4x: expanding gives 6x + 10 − 4, which simplifies to 6x + 6 = 4x. Subtract 4x from both sides: 2x + 6 = 0. Subtract 6: 2x = −6. Divide by 2: x = −3. Expand, collect, rearrange, divide.

Move the smaller x term across by subtracting it from both sides, then move the numbers the other way. For 4x + 3 = 2x + 15: subtract 2x to get 2x + 3 = 15, subtract 3 to get 2x = 12, divide by 2 to get x = 6. American sites say “variables on both sides”; it is the same method.

Put your value back into the original equation and work out each side. If both sides give the same number, the solution is right. For 4x + 3 = 2x + 15 with x = 6: the left side is 4 × 6 + 3 = 27 and the right side is 2 × 6 + 15 = 27, so x = 6 is correct.