Equation: What It Is and How to Solve One

What an Equation Is

An equation, according to the entry “Equazione” in the Treccani Encyclopedia of Mathematics, is an equality between two expressions containing one or more variables, called unknowns. Solving it means finding the values of the unknown that make that equality true. Equations are one of the objects that algebra works with: literal calculation, meaning the use of letters in place of numbers, is the tool used to write and manipulate them.

To move from one equation to another, simpler one with the same solutions, two rules are used. Treccani calls them equivalence principles. The first says that you can add (or subtract) the same expression on both sides of an equation, and the result is still an equivalent equation, meaning one with the same set of solutions. The second says that you can multiply (or divide) both sides by the same expression, as long as it’s never zero, again getting an equivalent equation.

The idea of equivalence is broader than it looks at first glance. Treccani, in its entry “Equazioni equivalenti,” gives a useful example: the equations |x|-1=0 and x²-1=0 both have -1 and +1 as solutions, so they’re equivalent, even though they aren’t the same equation at all and one can’t be obtained from the other using the two principles above.

First-Degree Equations, Step by Step

A first-degree equation in one unknown has the form ax+b=0, where a and b are known numbers and x is the unknown, with a not equal to zero. Solving it means applying the two equivalence principles until x is isolated by itself on one side of the equals sign, using the same addition, subtraction, multiplication and division you already meet in arithmetic.

Practical example: buying 3 identical notebooks, with a 2-dollar discount off the total, and spending 13 dollars in total, the price x of one notebook is found by solving the equation 3x - 2 = 13. By the first equivalence principle, add 2 dollars to both sides: 3x = 15. By the second principle, divide both sides by 3: x = 5. Each notebook costs 5 dollars.

Equations with Fractions

When the unknown appears inside fractions, as in x/2 + 1 = 5/3, the first step isn’t to calculate right away but to clear the denominators. You multiply both sides by the least common multiple of the denominators that appear in the equation: in this case, the least common multiple of 2 and 3 is 6. Multiplying every term by 6 makes the fractions disappear, leaving an ordinary first-degree equation, which is solved with the steps shown above. A common mistake is skipping this step and trying to add the fractions as they are: that also works, but the arithmetic gets much messier much faster once the denominators aren’t as simple as 2 and 3.

Second-Degree Equations and the Quadratic Formula

A second-degree equation has the general form ax²+bx+c=0, with a not equal to zero: compared to a first-degree equation, it also has an x² term, meaning the unknown squared. Wolfram MathWorld gives the formula for its solutions: x = (-b ± √(b²-4ac)) / 2a.

The expression under the square root, b²-4ac, is called the discriminant, and on its own it tells you how many real solutions exist before you even finish the calculation. If it’s positive, there are two distinct real solutions. If it’s zero, the formula produces a single solution, called a double root. If it’s negative, the square root of a negative number doesn’t exist among the real numbers, so the equation has no real solutions.

The history of this formula is longer and less straightforward than a single name might suggest. According to MacTutor History of Mathematics, the Babylonians, around 1800 BC, already solved special cases of quadratic equations with an algorithmic approach based on completing the square, even without having the concept of an equation itself; Euclid, around 300 BC, reached equivalent results by geometric means. It was the Persian mathematician al-Khwarizmi, in a treatise from around 800 from which the terms “algebra” and “algorithm” derive, who systematically classified several types of second-degree equations and gave, for each one, a solving rule paired with a geometric proof: work that MacTutor describes as the first organized treatment of the subject, rather than an invention out of nowhere.

Inequalities: Same Logic, a Sign That Can Flip

An inequality, according to Treccani, is a formula in which one of the inequality symbols appears — less than (<), less than or equal to (≤), greater than (>), greater than or equal to (≥) — in place of the equals sign. Otherwise, it’s solved by applying the same two equivalence principles used for equations, with one exception.

Treccani gives a direct example of this flip: from the inequality -2x > 3, multiplying both sides by -1/2 gives x < -3/2. The direction switched from greater-than to less-than exactly because the number used to multiply, -1/2, is negative.

There’s also a small difference worth noting when the inequality is “wide,” meaning it uses ≤ or ≥ instead of < or >: in that case, as Treccani observes, the set of solutions of the inequality also includes any solutions of the associated equation, obtained by replacing the inequality symbol with an equals sign. When the unknown ends up under a square root, the method changes again: inequalities with square roots also require an existence condition on the radicand before the two sides can be compared. When the unknown appears in a denominator instead, rational inequalities call for studying the sign of the numerator and denominator separately with a sign chart.

When there’s more than one unknown, a system of equations or inequalities to solve together enters the territory of linear algebra, which handles these systems with matrices and vectors. Those who continue their studies later meet equations involving continuous functions in calculus, where they’re also solved with the help of limits and derivatives. When the unknown sits in the exponent instead, the two equivalence principles seen here stop being enough: that’s the territory of exponential equations, which take a different method depending on whether the bases match or not.