From simple one-step equations to complex systems — learn to solve them all with confidence.
An equation is a mathematical statement that two expressions are equal, connected by the "=" sign. Solving an equation means finding all values of the variable(s) that make the statement true.
The fundamental principle of equation solving: whatever you do to one side, you must do to the other. This preserves the equality while isolating the unknown.
A linear equation in one variable has the form ax + b = c, where the variable x appears only to the first power. The graph of a linear equation in two variables is always a straight line (hence the name).
Step 1: Add 7 to both sides: 4x = 20
Step 2: Divide both sides by 4: x = 5
Check: 4(5) − 7 = 20 − 7 = 13 ✓
Step 1: Subtract 2x from both sides: 3x + 3 = 18
Step 2: Subtract 3: 3x = 15
Step 3: Divide by 3: x = 5
When an equation contains fractions, multiply every term by the least common denominator (LCD) to clear the fractions first.
LCD = 12. Multiply every term by 12:
4x + 3x = 84
7x = 84 → x = 12
Linear equations connect directly to linear functions, whose graphs are straight lines with slope m and y-intercept b in the form y = mx + b.
A quadratic equation has the standard form ax² + bx + c = 0 (where a ≠ 0). These equations can have 0, 1, or 2 real solutions.
If you can factor the quadratic, set each factor equal to zero (see the Polynomials & Factoring page for more techniques).
Factor: (x − 2)(x − 3) = 0
x − 2 = 0 → x = 2 or x − 3 = 0 → x = 3
The expression Δ = b² − 4ac is called the discriminant. It tells you the nature of the solutions:
a = 2, b = 3, c = −5
Δ = 9 − 4(2)(−5) = 9 + 40 = 49
x = (−3 ± 7) / 4
x = 1 or x = −5/2
This technique rewrites ax² + bx + c as a(x − h)² + k, revealing the vertex of the parabola.
x² + 6x = −2
x² + 6x + 9 = −2 + 9 (add (6/2)² = 9 to both sides)
(x + 3)² = 7
x = −3 ± √7
A system of equations is a set of two or more equations with the same variables. The solution is the set of values that satisfies all equations simultaneously.
y = 2x + 1
3x + y = 11
Substitute: 3x + (2x + 1) = 11 → 5x = 10 → x = 2, y = 5
2x + 3y = 12
4x − 3y = 6
Add the equations: 6x = 18 → x = 3, then y = 2
For larger systems, matrix methods are far more efficient. See Linear Algebra for Gaussian elimination, and the formula sheet for Cramer's Rule.
The absolute value |x| gives the distance of x from zero. To solve |expression| = k (where k ≥ 0), split into two cases:
Case 1: 2x − 5 = 9 → x = 7
Case 2: 2x − 5 = −9 → x = −2
Solution: x = 7 or x = −2
A radical equation contains a variable inside a radical (√). Isolate the radical and square both sides — but always check for extraneous solutions!
Square both sides: x + 3 = x² − 2x + 1
Rearrange: x² − 3x − 2 = 0
Factor/quadratic formula: x = (3 ± √17)/2
Check both in the original equation — reject any that produce a false statement.