Equations

An equation states that two expressions are equal. Solving an equation means finding the values of its variables that make the equality true, within the allowed domain.

For example, \(x+2=7\) is true when \(x=5\). Substituting that value into the original equation gives \(5+2=7\), confirming the solution.

Equations and expressions

An expression such as \(3x+1\) represents a quantity. An equation such as \(3x+1=10\) asserts an equality. The equals sign connects the two sides; it is not an instruction to perform an unrelated next step.

Some equations have one solution, some have several, and some have none. An identity is true for every value in its stated domain, such as \(2(x+1)=2x+2\) for real x.

Linear equations

A linear equation in one variable can be written as \(ax+b=0\), with \(a\ne0\). To solve it, perform equivalent operations on both sides.

For example, solve \(3x-4=11\):

\[3x=15,\qquad x=5.\]

We added 4 to both sides, then divided both sides by 3. The check is \(3(5)-4=11\).

Linear equations can also involve more than one variable, as in \(2x+y=7\). A single equation of this kind generally describes many ordered pairs rather than a unique pair.

Quadratic and cubic equations

A polynomial equation’s degree is its highest variable exponent after simplification, provided the leading coefficient is nonzero.

A quadratic equation has degree 2: \(ax^2+bx+c=0\), with \(a\ne0\).
A cubic equation has degree 3: \(ax^3+bx^2+cx+d=0\), with \(a\ne0\).

For instance, \(x^2-5x+6=0\) factors as

\[(x-2)(x-3)=0.\]

A product of real numbers equals zero only if at least one factor is zero. The solutions are \(x=2\) and \(x=3\). For quadratics that do not factor conveniently, use the quadratic formula.

A simple cubic, \(x^3=8\), has the real solution \(x=2\). Cubic equations are not generally solved by taking logarithms.

Exponential equations

In an exponential equation, the unknown appears in an exponent. For example, \(5^x=5^3\) is exponential, not cubic. Since a positive base other than 1 gives a one-to-one exponential function, matching the exponents gives \(x=3\).

When the bases do not match conveniently, logarithms can help. For \(2^x=7\),

\[x=\frac{\ln7}{\ln2}\approx2.8074.\]

By contrast, \(x^3=7\) has a variable base and a fixed exponent; it is a polynomial equation. The position of the variable matters. See our introduction to logarithms.

Systems of equations

A solution of a system must satisfy every equation in the system. Consider

\[y=3x-2,\qquad y=-x-6.\]

Substitution gives \(3x-2=-x-6\), so \(4x=-4\) and \(x=-1\). Substituting back gives \(y=-5\). The ordered pair \((-1,-5)\) satisfies both original equations.

A graph shows this solution as the intersection of the two lines, but drawing a graph is not required: substitution and elimination can solve systems algebraically. Some systems have no solution, while others have infinitely many.

Operations that need extra care

Division: do not divide by zero or by an expression that could be zero without considering that case separately.
Squaring: squaring both sides can introduce extra candidates. For \(x=-2\), squaring gives \(x^2=4\), but the candidate \(x=2\) does not solve the original equation.
Square roots: solving \(x^2=9\) gives both \(x=3\) and \(x=-3\).
Domain restrictions: real square roots require nonnegative radicands, and real logarithms require positive arguments.

Finish by checking candidates in the original equation. That check is especially important after an operation that may not preserve equivalence in both directions.