Solving Simple Equations

An equation is a statement that two numbers or number expressions are equal, like 2 = 5 - 3 or 4x = 2x + 12. The part to the left of the = sign is the first or left member, and the part to the right is the second or right member. A letter whose value we're looking for is called the unknown.

An axiom is an evident truth accepted without proof. Solving equations uses four of them constantly:

1. Adding the same number to both members keeps the equation true.
2. Subtracting the same number from both members keeps it true.
3. Multiplying both members by the same number keeps it true.
4. Dividing both members by the same number (not zero) keeps it true.

An equation as simple as 3x = 12 is solved by dividing both members by 3, the coefficient of x, giving x = 4. If all the unknown terms are already on one side and all the plain numbers on the other, combine like terms first: 5x - 2x + 2x = 8 + 15 - 3 becomes 5x = 20, so x = 4.

Moving terms across

Usually both members have unknown terms and number terms, as in 5x - 4 = 2x + 17. The axioms let us reshape it step by step until it looks like 3x = 12:

\begin{array}{rll} 5x - 4 &= 2x + 17 & \\ 3x - 4 &= 17 & \text{subtract } 2x \text{ from each member (Axiom 2)} \\ 3x &= 21 & \text{add 4 to each member (Axiom 1)} \\ x &= 7 & \text{divide each member by 3 (Axiom 4)} \end{array}

A simple or linear equation is one that can be written so that each unknown appears only to the first power, no term has two unknowns multiplied together, and no unknown sits in an exponent or a denominator. So 5x - 2 = 8 and x - 2y + 3 = 6 are simple, while x^2 = 4 and x - xy = 8 are not.

Checking a solution

Rule. Substitute the value you found for the unknown in the original equation, then simplify until the two members are plainly identical.

Check x = 7 in 5x - 4 = 2x + 17: \;5 \cdot 7 - 4 = 2 \cdot 7 + 17, so 35 - 4 = 14 + 17, or 31 = 31. It checks.